UNIVERSITY  OF  CALIFORNIA 

MEDICAL  CENTER  LIBRARY 

SAN  FRANCISCO 


Dr.  Howard  I.  Hawdsley 


\. 


A  MANUAL 


OF 


PHYSICAL  MEASUREMENTS 


BY 


JOHN  O.  [R_EED,  PH.D. 

PROFESSOR   OF   PHYSICS   IN   THE   UNIVERSITY    OF   MICHIGAN 

AND 

KARL  B.  GUTHE,  PH.D. 

PROFESSOR   OF   PHYSICS   IN   THE   UNIVERSITY    OF   MICHIGAN 


FOURTH  EDITION,  REVISED  AND  ENLARGED 


QC37 

R3L 
l°\\3 


GEORGE  WAHR,  PUBLISHER 
ANN   ARBOR,   MICHIGAN 

1913 


COPYRIGHT,  1902 

BY 
JOHN  O.  REED 

AND 
KARL  E.  GUTHE 


COPYRIGHT,  1906 

BY 
JOHN  O.  REED 

AND 

KARL  E.  GUTHE 

COPYRIGHT,  1912 

BY 
JOHN  O.  REED 

AND 
KARL  E.  GUTHE 


THE  ANN   ARBOR    PRESS 


PREFACE. 


This  manual  has  been  prepared  to  meet  the  needs  of  students 
beginning  work  in  the  Physical  Laboratory  of  the  University  of 
Michigan.  Such  a  book  must  inevitably  possess  a  certain  local 
coloring  peculiar  to  the  conditions  it  has  been  designed  to  meet. 
A  manual  equally  suited  to  all  laboratories,  has  not  been  and 
probably  will  not  be  written.  Each  laboratory  reflects  in  greater 
or  less  degree  the  individual  trend  of  the  man  who  stands  at 
its  head;  and  its  exercises  and  methods  are  the  result  of  an  ex- 
tended process  of  adaptation  and  assimilation.  Hence  it  happens 
that  one  laboratory  is  largely  devoted  to  the  study  of  the  phe- 
nomena of  light,  another  to  those  of  electricity,  and  a  third  to 
those  of  elasticity,  heat,  or  electrochemistry,  as  the  case  may  be. 
The  moral  of  all  this  is,  that  the  practice  and  traditions  of  each 
laboratory  are  best  conserved  by  a  text  representative  of  its  own 
methods,  and  if  no  better  reason  should  be  found,  perhaps  this 
may  serve  to  explain  the  appearance  of  this,  another  laboratory 
manual. 

The  exercises  herein  described  embody  the  work  required  of 
students  in  Physics  and  in  Engineering  in  their  first  course  in 
Physical  Laboratory  Practice.  Such  a  course  is  expected  to  oc- 
cupy three  laboratory  periods  of  two  hours  each  for  one  semester, 
and  embraces  some  thirty-six  to  forty  of  the  exercises  in  this 
manual.  Owing  to  the  diversity  of  the  work  prescribed  in  the 
various  courses  in  Engineering,  no  one  student  is  expected  to 
complete  all  the  exercises  in  this  book  in  a  single  semester. 

In  accordance  with  the  practice  in  the  University  of  Michigan, 
it  is  expected  that  the  laboratory  work  shall  be  supplemented  by 
lectures  upon  the  theory  of  the  exercises,  and  recitations  upon  the 
work  actually  done  and  the  results  obtained.  In  this  way  it  is 
believed  that  the  student  is  brought  to  a  clearer  understanding  of 
the  significance  of  the  exercise  and  of  the  accuracy  attainable 


vi  PHYSICAL    MEASUREMENTS 

under  given  conditions.  To  this  end  the  exercises  are  numbered 
consecutively  throughout  the  text,  and  those  under  any  specific 
subject  are  preceded  by  sufficient  theory  to  render  the  formulae 
and  methods  clear  to  persons  familiar  with  the  fundamental 
principles  of  Physics  as  set  forth  in  any  standard  textbook. 

Being  designed  for  beginners  in  the  Physical  Laboratory,  this 
manual  makes  no  claim  to  completeness,  either  in  subject  mat- 
ter or  in  exposition.  The  aim  has  been  to  furnish  a  coherent  and 
logical  series  of  graded  exercises  in  Physical  Measurement,  such 
as  will  best  furnish  an  introduction  to  Practical  Physics,  and  at 
the  same  time  afford  opportunity  for  developing  ability  in  record- 
ing and  interpreting  observations,  and  skill  in  the  manipulation 
of  delicate  and  sensitive  apparatus. 

For  convenience  of  reference  a  series  of  tables  of  the  more 
important  physical  constants,  of  squares,  cubes,  square  roots  and 
multiples  of  TT,  of  the  logarithms  of  numbers,  and  the  trigonomet- 
ric functions  have  been  added.  A  thorough  drill  in  the  use  of 
logarithmic  tables  in  the  computation  of  results,  should  form  a 
feature  of  any  successful  course  in  Laboratory  Practice.  To  this 
end  an  orderly  method  of  procedure  in  such  computation  has  at 
all  times  been  insisted  upon. 

The  authors  have  drawn  freely  from  many  standard  works  on 
Practical  Physics,  notably  .from  those  of  Kohlrausch,  and  Stew- 
art and  Gee  in  General  Physics,  and  from  Carhart  and  Patter- 
son's Electrical  Measurements. 

In  conclusion  we  wish  to  thank  our  colleagues,  Professors 
Carhart  and  Patterson,  for  helpful  suggestions  and  criticisms 
during  the  preparation  of  the  work. 

University  of  Michigan,  March,  1902. 


FROM  THE  PREFACE  TO  THE  SECOND  EDITION. 


The  necessity  for  a  second  edition  of  this  book  has  presented 
an  opportunity  for  a  careful  revision  of  the  text,  both  in  the 
elimination  of  errors  and  in  the  addition  of  certain  features  which 
experience  has  shown  to  be  desirable  and  necessary  to  make  the 
manual  truly  representative  of  modern  laboratory  practice.  In 
making  these  additions  the  needs  of  the  average  student  have 
been  kept  constantly  in  mind  both  as  regards  his  previous  prep- 
aration and  the  requirements  laid  upon  him  by  his  subsequent 
University  work. 

In  the  first  part  several  articles  are  devoted  to  the  measure- 
ment of  angles,  and  a  chapter  has  been  added  upon  Surface 
Tension  and  Viscosity.  In  order  to  meet  more  fully  the  demands 
made  upon  students  of  Mechanical  and  Electrical  Engineering 
the  chapters  upon  Heat  and  Electricity  have  been  practically 
rewritten.  Several  of  the  articles  in  these  chapters  contain  new 
and  important  matter,  notable  among  which  are  the  discussion  of 
the  ballistic  d'Arsonval  galvanometer,  and  the  exercises  involving 
the  use  of  the  potentiometer  and  of  the  thermoelement. 

While  the  additions  have  in  general  been  such  as  to  render  the 
work  more  advanced  in  character,  with  the  possible  exception  of 
some  exercises  in  the  measurement  of  angles,  still  it  is  hoped 
that  the  book  will  not  be  found  less  useful  for  elementary  work 
than  before.  The  forms  for  recording  results  and  the  outlines 
for  computation  have  abundantly  justified  the  wisdom  of  their 
insertion  in  the  immense  saving  of  time  and  energy  to  the  busy 
instructor.  While  it  has  been  urged  by  some  that  students  readily 
and  intuitively  devise  explicit,  symmetrical  and  logical  arrange- 
ments for  their  data  and  computations,  such  students  have  as  yet 
entirely  escaped  our  observation. 

September,  1906. 


PREFACE  TO  THE  THIRD  EDITION. 


The  third  edition  of  this  manual  has  been  prepared  mainly 
"because  it  was  felt  that  the  arrangement  of  the  subject  matter 
should  correspond  more  closely  to  that  found  in  the  authors' 
COLLEGE;  PHYSICS,  recently  published  by  the  Macmillan  Company. 

The  book  has  also  been  thoroughly  revised,  the  treatment  been 
changed  in  a  number  of  places  and  a  few  exercises,  notably  some 
elementary  exercises  in  electricity,  .been  added. 

The  authors  are  greatly  indebted  to  their  colleagues,  Professor 
H.  M.  Randall  and  Mr.  W.  W.  Sleator,  for  assistance  in  reading 
the  proof  sheets  of  the  present  edition. 

JOHN  O.  RSED. 
KARL  E.  GUTHE. 

August,  1912. 


TABLE  OF  CONTENTS. 


INTRODUCTION. 

ARTICLE.  PAG£- 

1  Benefits  of  laboratory  work I 

2  Instruments    I 

3  Record  of  observations   .' .  2- 

4  Graphical   methods    2 

5  Errors  of  observation 5 

6  Probable  error   6 

7  Influence  of  errors  upon  the  result 7 

8  Interpolation    9- 

9  Hints  on  computation 9- 


CHAPTER  I. 

FUNDAMENTAL  MEASUREMENTS. 
10    Fundamental  magnitudes  II 


LENGTH. 

1 1  Contact    measurements    1 1 

12  Methods  of  subdivision   12 

13  The  micrometer  screw  12 

14  Exercise  i.    The  micrometer  gauge 12 

15  Exercise  2.    The  'spherometer 13 

16  The    vernier    15 

17  Exercise  3.    The  vernier 15 

18  Exercise  4.    The  vernier  caliper  16- 

19  Line    measurements    17 

20  The  cathetoimeter  17 

21  Optical  micrometers    17 

22  The  dividing  engine   21 

23  'Exercise  5.    The  dividing  engine 22 : 


X  PHYSICAL    MEASUREMENTS 

ANGLE. 
ARTICLE.  PAGE. 

24  Measurement   of   angle 23 

25  Definitions    23 

26  Exercise  6.    The  protractor  24 

27  Exercise  7.    Angles  from  trigonometric  functions 25 

28  Correction   for  eccentricity    26 

29  Telescope  and  scale   27 

30  Exercise  8.     The  optical  lever  28 

31  The  lever  tester 30 

32  Exercise  9.    Constants  of  the  level 31 

33  Small  angles  'by  the  filar  micrometer 33 

34  Exercise  10.    Angular  measurement  by  the  filar  micrometer 34 

35  The  sextant  34 

36  Exercise  n.    Measurement  of  angles  by  the  sextant. 37 


MASS. 

37  The   balance    37 

38  Determination  of  the  resting  point   38 

39  Sensibility  of  the  balance 39 

40  Exercise  12.    To  make  a  single  weighing 39 

41  Reduction  to  weight  in  vacuum 41 

42  Exercise  13.     Double  weighing 41 


TIME. 

43  Period  of  vibration 42 

44  Method  of  coincidences  43 

45  Exercise  14.    Period  of  torsional  pendulum 47 

46  Exercise  15.    The  barometer  47 


CHAPTER  II. 

ELASTICITY. 

pAGE 

47  Definitions    

48  Hooke's  law .....!!.!..!..!..!!.!...!  40 

49  Coefficients  of  elasticity 50 

50  Coefficient  of  volume  elasticity so 


CONTENTS 


XI 


ARTICLE. 
51 

52 

53 

54 
55 
56 

57 


59 


PAGS. 

Young's    modulus    51 

Simple  rigidity   51 


Exercise  16.    To  verify  Boyle's  law 54 

Exercise  17.     The  Jolly  balance  57 

Exercise  18.     Young's  modulus  by  stretching 59 

Exercise  19.    Verification  of  the  laws  of  bending 60 

Exercise  20.     Young's  modulus  by  flexure  66 

Exercise  21.    (Simple    rigidity    67 

Exercise  22.  Simple    rigidity   of    a   brass    wire    from   torsional 

vibrations    69 


CHAPTER  III. 


PENDULUM  EXPERIMENTS  AND  MiOMENT  OF  INERTIA. 

60  The  simple  pendulum  71 

61  Exercise  23.     Law  of  the  simple  pendulum 71 

62  Exercise  24.     Computation   of   g 73 

63  Exercise  25.     Moment  of  inertia  of  a  connecting  rod 73 

64  Exercise  26.     Moment  of  inertia  from  torsional  vibrations 75 

65  Exercise  27.     The   ballistic   pendulum 78 


CHAPTER  IV. 

DENSITY. 

66.     Definition    81 

67  Exercise  28.     Density  from  mass  and  volume 81 

68  Exercise  29.    The  pyknometer   81 

69  Exercise  30.     Mohr's  balance    82 


CHAPTER  V. 


SURFACE  TENSION  AND  VISCOSITY. 

70  Characteristics  of  a  liquid   84 

71  Exercise  31.     Measurement  of   surface  tension 84 

72  Exercise  32.     Surface  tension   from  capillary  action 86 

73  Coefficient    of    viscosity 87 

74  Exercise  33.     Coefficient  of  viscosity  by  flow  through  a  capillary 

tube   .  ..88 


xii  PHYSICAL    MEASUREMENTS 

CHAPTER  VI. 

MEASUREMENTS  IN  SOUND. 

75  Exercise  34.    Velocity  of  sound  in  metals  (Kundt's  method) 91 

76  Exercise  35.     Computation  of  Young's  modulus 93 

77  Exercise  36.    Rating  a  turning  fork.     Graphical  method 93 


CHAPTER  VII. 
MEASUREMENTS  IN  HEAT. 

ARTICLE. 

78  Effects  of  heat  96 

THERMOMETRY. 

79  Thermometry  96 

80  Exercise  37.    Determination  of  the  fixed  points  of  a  thermometer  97 

81  -Stem  correction   100 

EXPANSION. 

82  Coefficient  of  linear  expansion 100 

83  Exercise  38.    Coefficient  of  linear  expansion  of  a  solid 101 

84  Expansion  of  liquids    104 

85  Exercise  39.    Coefficient  of  expansion  of  a  liquid  by  the  dilato- 

meter    106 

86  Air  free  water 109 

87  Exercise  40.     Constant  volume  air  thermometer 109 

CAI,ORIMETRY. 

88  Definitions 112 

89  Specific  heat  by  method  of  mixtures   113 

90  Exercise  41.    Water  equivalent  of  a  calorimeter   114 

91  Exercise  42.     Specific  heat  of  copper   116 

92  Correction  for  radiation   117 

93  Exercise  43.     Heat  of  fusion  of  water  120 

94  Exercise  44.    Heat  of  vaporization  of  water  at  boiling  point 121 

95  Exercise  45.     Melting  point  and  heat  of  fusion  of  tin   124 

VAPOR  PRESSURE. 

96  Measurement  of  vapor  tension    126 

97  Exercise  46.     Vapor  tension  of  ether   127 

98  Exercise  47.    Vapor  tension  of  water  at  various  temperatures.  .  127 


CONTENTS  X1H 

CHAPTER  VIII. 
ELECTRICAL  MEASUREMENTS. 

UNITS  AND  STANDARDS. 
ARTICLE.  PAGE. 

99    Resistance   130 

100  Current    134 

101  Electromotive   force    134 

102  Quantity  o"f  electricity  136 

103  Capacity    136 

104  Selfinductance    137 

INSTRUMENTS. 

105  Keys     138 

106  Galvanometers     139 

107  The  astatic  galvanometer   140 

108  The  d'Arsonval  galvanometer  140 

109  Methods  of  observation  142 

1 10  Shunts    142 

in     Exercise  48.     Calibration  of  a  galvanometer  by  Ohm's  law 143 

112  Exercise  49.     Figure  of  merit  of  a  galvanometer   144 

1 13  Ballistic    galvanometers    147 

1 14  Constant  of  ballistic  galvanometer   148 

115  Exercise  50.     Determination   of  the  constant  of  a  ballistic  gal- 

vanometer   148 

1 16  Voltmeters  and  ammeters    149 

ELEMENTARY  EXERCISES. 

117  Exercise  51.     Cells  in  series  and  in  parallel  150 

118  Exercise  52.     Kirchhoff's  laws   151 

119  Exercise  53.     Resistances  in  series  and  in  parallel 153 

MEASUREMENT  O?  RESISTANCE. 

120  Exercise  54.    Resistance   by   substitution    1,54 

121  Exercise  55.     Resistance  by  voltmeter  and  ammeter 15,5 

122  Exercise  56.     Very  high  resistances  by  direct  deflection 156 

123  The  Wheatstone  bridge    157 

124  Exercise  57.     Resistance  by  Wheatstone  bridge  box 158 

125  Exercise  58.     Resistance  by  slide-wire  bridge  160 

126  Exercise  59.     Resistance  of  a  galvanometer — Thomson's  method  162 

127  Exercise  60.     Resistance  of  a  galvanometer — Second  method....  163 

128  Exercise  61.     Resistance   of   an   electrolyte 164 


XIV  PHYSICAL    MEASUREMENTS 

ELECTROMOTIVE  FORCE  AND   POTENTIAL  DIFFERENCE. 

ARTICLE. 

129  The  voltmeter    167 

130  Exercise  62.    Electromotive  force  of  a  cell  167 

131  Exercise  63.    Electromotive   force  by  potentiometer  method....  168 

132  The   potentiometer    170 

133  Exercise  64.     Calibration  of  a  voltmeter  171 

134  Exercise  65.    Thermo-electromotive  force  of  a  thermoelement..   172 

ELECTROMOTIVE  FORCE  AND  RESISTANCE  OF  BATTERIES. 

135  Electromotive  force  and  difference  of  potential  173 

136  Exercise  66.    Terminal    potential    difference    as    a    function    of 

external   resistance    175 

137  Exercise  67.    Electromotive  force  and  internal  resistance,  volt- 

meter and  ammeter   176 

138  Exercise  68.     Electromotive    force    and    internal    resistance    by 

condenser  method    177 

139  Exercise  69.     Internal  resistance  by  method  of  Nernst  and  Haagn  179 

MEASUREMENT  OF  CURRENT. 

140  Measurable  effects  of  a  current 181 

141  Law  of  electrolysis  181 

142  The  copper  coulometer   •. 182 

143  Exercise  70.     Calibration    of    an   instrument   by   use    of    copper 

coulometer    182 

144  Exercise  71.     Calibration  of  ammeter  by  standard  cell  183 

COMPARISON  OF  CAPACITIES. 

145  Exercise  72.    Comparison  by  direct  deflection  184 

146  Exercise  73.    Method  of  mixtures   186 

MEASUREMENT  OF  INDUCTANCE. 

147  Exercise  74.    iSelfinductance  of  a  coil  compared  with  a  standard  187 

148  Exercise  75.    Mutual  inductance  of  two  coils   .  .   189 


CHAPTER  IX. 
MAGNETIC  MEASUREMENTS. 


MAGNETIC  FIELDS. 

149  Magnetic  fields  192 

150  Exercise  76.     Determination  of  H  (First  method)  193 

151  Exercise  77.    Determination  of  H  (Second  method)    198 


CONTENTS  XV 

MAGNETIC  PROPERTIES  OF  IRON  AND  STEEL. 

152  Magnetic  permeability   200 

153  Exercise  78.    Commutation  curve  for  iron  and  steel 201 

154  Exercise  79.     Hysteresis  curve  for  iron  and  steel 205 


CHAPTER  X. 

OPTICAL  MEASUREMENTS. 

CURVATURE. 

PAGE. 

155  Curvature  of  optical  surfaces 208 

156  Exercise  80.    Radius  of  curvature  of  a  lens  by  the  spherometer.  208 

157  Exercise  81.    Radius  of  curvature  by  reflection 210 

158  Exercise  82.    Focal  length  of  lenses  213 

159  Exercise  83.    !Lens  curves   215 

MAGNIFYING    POWER. 

160  Exercise  84.    Magnifying  power  of  the  telescope  216 

161  Exercise  85.     Magnifying  power  of  microscope  218 

INDEX  OF  REFRACTION. 

162  Exercise  86.     Index  of  refraction  of  lenses  from  radii  of  curva- 

ture and  focal  lengths 219 

163  Exercise  87.    Index  of  refraction  by  means  of  a  microscope....  220 

THE   SPECTROMETER. 

164  Description     222 

165  Adjustments  of  the  spectrometer 224 

166  Reflecting  surfaces 226 

167  Exercise  88.     To  measure  the  angle  of  a  prism  228 

168  Angles  by  method  of  repetition  230 

169  Exercise  89.     Index  of  refraction  of  a  glass  prism 232 

DIFFRACTION. 

i/o    Exercise  90.     Wave  lengths  of  sodium  light  by  diffraction  grat- 
ing      234 

171  Exercise  91.     Constant  of  a  diffraction  grating  235 

172  Dispersion,  normal  and  prismatic   236 

173  Exercise  92.     Dispersion  curve  for  a  prism  237 


Xvi  PHYSICAL    MEASUREMENTS 

TABLES. 

I.  Atomic  weights  of  some  elements  239 

II.  Density  of  water  at  different  temperatures  239 

III.  Density  of  mercury  at  different  temperatures  240 

IV.  Density  of  various  bodies    240 

V.  Reduction  of  barometer  readings  to  o°C 241 

VI.  Coefficients  of  elasticity  241 

VII.  Viscosity  and  surface  tension  of  liquids  at  2O°C 242 

VIII.  -Moments  of  inertia  242 

IX.  Boiling  point  of  water  under  different  barometric  pressures..  243 

X.  Heat  constants    243 

XI.  Vapor  tension  of  liquids   244 

XII.  Index  of  refraction  for  sodium  light  244 

XIII.  Wave  lengths  of  lines  in  solar  spectrum 244 

XIV.  Electrical  resistance  of  metals  245 

XV.  Electrical  conductivity  of  solutions  at  i8°C 245 

XVI.  Numbers  frequently  required    246 

XVII.  Numerical    tables    247,  248 

XVIII.  Trigonometric    functions    249,  250 

XIX.  Logarithms    ; 251,  252 


<J  ( 


PHYSICAL  MEASUREMENTS 


INTRODUCTION 

1.  Benefits  of  Laboratory  Work.    The  benefits  to  be  derived 
by  the  student  from  work  in  the  physical  laboratory  are  two-fold. 
In  the  first  place  he  is  to  become  acquainted  with  delicate  instru- 
ments, to  be  trained  to  make  systematic,  accurate  and  independent 
observations,  and  to  compute  from  the  data  so  obtained,  the  val- 
ues of  many  of  the  more  important  physical  constants.   Secondly, 
he  is  to  make  a  searching  review  of  the  fundamental  principles  of 
the  science,  and  to  be  brought  to  a  more  lively  realization  of  the 
meaning  and  importance  of  formulae  and  laws  deduced  in  the 
text-books   upon   general   physics.     To   this   end   it   is   urgently 
advised   that   the   student   familiarize   himself   with   all   the   de- 
tails of  the  theory  of  the  experiment,  and  be  able  to  sketch  from 
memory  the  apparatus  to  be  employed  before  beginning  any  ex- 
periment.   An  attempt  to  follow  directions  but  dimly  understood, 
and  to  manipulate  apparatus  whose  construction  and  purpose  are 
alike  unknown,  can  only  result  in  loss  of  time,  and  laboratory 
work  under  such  circumstances  is  practically  worthless  as  a  means 
of  discipline. 

2.  Instruments.     The  instruments  used  in  the  physical  lab- 
oratory are  usually  of  delicate  construction;  many  of  them  are 
costly  and  liable  to  injury  from  rough  or  careless  usage.    It  is  of 
the  highest  importance  that  all  apparatus  should  be  handled  with 
care,  and  returned  to  its  proper  place  after  use.     If  any  piece  is 
found  to  be  out  of  adjustment  or  in  need  of  repair,  report  the  fact 


2  PHYSICAL    MEASUREMENTS 

before  beginning  work.  If  any  screws  or  other  parts  of  an  instru- 
ment do  not  move  readily,  do  not  apply  force  but  report  the 
matter  to  the  instructor  in  charge.  The  ability  to  use  delicate 
apparatus  without  injuring  or  destroying  it  is  an  important  part 
of  a  liberal  education. 

3.  Record  of  Observations.    All  observations,  data  and  nec- 
essary formulae,  such  as  the  time  and  place  of  the  exercise,  the 
specific  instruments  used,  the  objects  measured,  etc.,  are  to  be 
recorded  in  a  note  book  provided  for  that  purpose.     Such  a  book 
should  have  fixed  leaves,  and  be  made  of  paper   suitable  for 
writing  with  ink.     The  record  is  to  be  made  at  the  time  of  the 
experiment.    It  must  contain  the  detailed  information  necessary, 
must  be  clearly  written  and  arranged  in  a  neat,  methodical  man- 
ner, so  that  one  familiar  with  the  experiment  may  readily  compre- 
hend what  has  been  done.    It  should  be  sufficiently  specific  to  be 
intelligible  after  the  circumstances  of  the  experiment  are  entirely 
forgotten. 

A  suitable  form  of  record  has  been  appended  to  each  exercise 
in  the  manual,  and  the  student  will  do  well  to  follow  these,  at 
least  until  he  is  able  to  arrange  the  material  for  himself.  Tabula- 
tion of  results  in  columns  adds  much  to  the  neatness  of  a  note 
book  and  materially  assists  in  the  detection  of  errors,  either  of 
record  or  of  observation.  It  is  suggested  that  the  note  books 
be  casually  inspected  at  each  exercise  if  possible,  by  the  instructor, 
in  his  rounds  in  the  laboratory. 

The  computation  of  results  should  generally  be  done  at  home, 
and  a  record  of  the  complete  experiment  be  handed  to  the  in- 
structor for  approval.  In  this  report  the  date,  number,  name  and 
object  of  the  exercise  should  precede  a  short  discussion,  deriva- 
tion of  formulae,  the  record  of  observations  and  the  computed 
results.  If  these  reports  be  written  on  loose  sheets  to  be  returned 
to  the  student  after  approval  by  the  instructor,  these  sheets  must 
be  kept,  ready  for  inspection,  in  a  suitable  binder.  A  separate 
page  should  always  be  begun  for  each  new  experiment. 

4.  Graphical  Methods.    It  is  frequently  of  interest  to  verify 
a  law.  stating  the  relation  that  is  known  to  exist  between  two 


INTRODUCTION  3 

quantities,  or  to  detect  and  determine  such  a  relation  where  it  is 
not  known.  In  all  such  cases  the  results  obtained  by  observation 
are  most  clearly  presented  to  the  eye  when  plotted  as  a  curve.  - 
In  the  application  of  the  method  it  is  customary  to  plot  values  of 
the  independent  variable  as  abscissae  and  those  of  the  dependent 
variable  as  ordinates.  In  all  cases  where  the  phenomenon  under 
investigation  is  continuous,  a  smooth  curve  sketched  through  the 
points  obtained,  may  be  assumed  to  represent  the  facts  better  than 
any  individual  observation.  The  graphical  method  has  the  addi- 
tional advantage  that  an  accidental  error  is  at  once  made  evident 
by  the  fact  that  the  point  so  obtained  departs  markedly  from 
the  curve. 

The  curve  most  readily  plotted  and  tested  is  the  straight  line, 
and  it  is  advisable  so  to  transform  the  assumed  relation  as  to 
render  the  plotting  of  a  straight  line  practicable. 

For  example,  suppose  it  were  desired  to  investigate  the  relation 
between  the  time  of  vibration  of  a  pendulum  and  its  length.  If 
we  assume  that  this  relation  may  be  expressed  by  an  algebraic 
function  of  the  form 

T  =  ar  (i) 

we  may  determine  the  constants  a  and  m  from  a  series  of  obser- 
vations. Passing  to  logarithms  we  have 

log  T  =  m  log  /  +  log  a  (2) 

This  is  clearly  of  the  form 

y  =  m  x  +  c  (3) 

and  is  therefore  the  equation  of  a  straight  line.  If  now  we  plot 
values  of  log  /  as  abscissae  and  the  corresponding  values  of  log  T 
as  ordinates,  we  may  decide  at  once  whether  such  a  relation  as  we 


4  PHYSICAL    MEASUREMENTS 

have  assumed  exists,  and  we  may  obtain  values  of  a  and  m  direct- 
ly from  the  curve. 

It  is  not  necessary  that  the  same  numerical  value  should  be 
assigned  to  a  scale  division  on  the  horizontal  and  vertical  axes. 
In  general  it  is  best  to  make  the  value  of  a  scale  division  cor- 
respond, as  nearly  as  possible,  to  the  least  quantity  which  we  can 
measure.  If  this  be  impossible,  such  values  should  be  chosen  for 
a  scale  division  on  each  axis  as  will  cause  the  curve  most  nearly 
to  fill  the  page  with  the  observations  to  be  plotted.  It  is  only  in 
case  we  wish  to  determine  from  the  curve  a  quantity  which  is 
represented  by  the  tangent  of  an  angle,  i.  e.,  which  represents  the 
ratio  between  the  coordinates,  that  it  is  necessary  to  assign  to 
them  their  proper  relative  values. 

A  table  of  the  data  from  which  the  curve  has  been  plotted 
should  in  all  cases  accompany  the  curve  and  the  values  as- 
signed to  a  scale  division  on  the  horizontal  and  vertical  axes 
must  be  clearly  stated  upon  these  axes.  In  practice  it  is  well 
to  prick  with  a  needle  the  exact  position  of  each  point  on  the 
curve  and  then  draw  around  each  point  a  small  circle  in  colored 
ink.  All  curves  should  be  plotted  upon  special  cross-section  paper, 
drawn  in  ink,  and  the  points  clearly  marked  as  indicated  above. 
In  case  special  accuracy  is  desired  it  is  well  to  use  paper  printed 
from  engraved  plates.  The  curves  are  to  be  inserted  in  the  note 
book  after  the  record  of  the  experiment.  This  is  most  readily 
done  by  cutting  away  about  two-thirds  of  a  sheet  lengthwise,  and 
pasting  the  stub  to  the  back  of  the  cross-section  paper. 

The  following  data  may  be  used  as  exercises  in  the  plotting  of 
curves.  In  case  any  set  of  data  does  not  represent  continuous 
phenomena  how  should  the  curve  be  drawn? 

I.  POPULATION  OF  THE  UNITED  STATES. 


Year 

1810 
1820 
1830 
1840 
1850 

Population 
3  929  214 
5  308  483 
7  239  881 
9  633  822 
12  866  020 
17  069  453 
23  191  876 

Year 
1860 
1870 
1880 
1890 
1900 
1910 

Population 
31  443  321 
38  558  371 
50  155  783 
62  622  250 
76  303  387 
93  402  151 

INTRODUCTION  5 

II.  ATTENDANCE  AT  THE  UNIVERSITY  OF  MICHIGAN. 

Collegiate  No.  of  Collegiate  No.  of 

Year  Students  Year  Students 

1881-82  1534  1897-98  3223 

1882-83  1440  1898-99  3192 

1883-84  1377  1899-1900  3441 

1884-85  1295  1900-1901  3712 

1885-86  1401  1901-1902  3709 

1886-87  15/2  1902-1903  3792 

1887-88  1667  1903-1904  3957 

1888-89  1882  1904-1905  4136 

1889-90  2153  1905-1906  4571 

1890-91  2420  1906-1907  4746 

1891-92  2692  1907-1908  5010 

1892-93  2778  1908-1909  5223 

1893-94  2659  1909-1910  5383 

1894-95  2864  1910-1911  5381 

1895-96  3014  1911-1912  5582 

1896-97  2975 

III.  HYSTERESIS  CURVE  FOR  SWEDISH  IRON. 

The  values  are  given  for  half  a  cycle  only.  To  obtain  the  com- 
plete curve,  reverse  the  signs  of  both  columns. 

Intensity  of  Field  Magnetic  Induction 

H  B 

o      —  8300 

1-5   —  5816 

2.7  o 

3-7   4-  3041 

5-0  4-  5284 

6.6  4-  7037 

8.8  4-    8658 

ii-3   4-    9923 

16.0  -j-  11389 

25.2 -f-  12898 

36.2  4-  13808 

45-9  +  14430 

62.4  4-  15074 

45-0  4-  14652 

21.1   4-  13653 

6.6  4-  11766 

4.6    -\-III22 

o     4-    8300 

5.  Errors  of  Observation.  In  any  series  of  measurements  of 
the  same  physical  quantity,  we  find  that  the  results  differ  slightly 
from  one  another,  owing  to  imperfections  of  the  instrument  or 
errors  in  making  the  readings.  These  errors  are  not  to  be  con- 


6  PHYSICAL    MEASUREMENTS 

founded  with  mistakes  in  calculation  or  errors  in  recording  obser- 
vations; such  errors  must  of  course  be  rejected.  First  to  be  con- 
sidered are  the  accidental  errors  of  observation  which,  if  the  obser- 
vations have  been  made  without  any  bias,  or  preconceived  idea 
as  to  what  the  value  ought  to  be,  are  likely  to  be  positive  as 
often  as  negative  and  may,  in  the  long  run,  be  considered  as 
having  little  influence  upon  the  mean  result.  Obviously  the  influ- 
ence of  such  errors  is  diminished  by  making  the  number  of  obser- 
vations as  large  as  possible  and  taking  the  arithmetical  mean. 
The  latter  may  be  regarded  as  an  observation  made  with  a  more 
precise  instrument.  The  average  of  n  observations  has  a  measure 
of  precision  ~\/n  times  as  large  as  that  of  a  single  observation. 
i.  e.,  if  we  take  the  mean  of  sixteen  observations,  then  the  prob- 
able error  of  this  mean  is  only  one-fourth  of  that  of  a  single 
observation. 

It  is  not,  however,  at  all  times  convenient  nor  necessary  to 
multiply  the  number  of  observations.  If  a  set  of  readings  differ 
but  little  among  themselves,  it  is  clear  that  little  will  be  gained 
by  increasing  the  number  of  such  observations.  On  the  other 
hand  if  the  individual  readings  differ  markedly  among  themselves, 
a  much  larger  number  must  be  taken  if  the  average  reading  is  to 
be  considered  as  trustworthy.  The  conditions  of  the  experiment 
and  the  degree  of  accuracy  desired,  must  in  all  cases  be  the  deter- 
mining factors. 

6.  Probable  Error.  What  is  more  to  the  purpose,  is  to  give 
a  clear  idea  of  the  degree  of  precision  attained  in  a  given  meas- 
urement. This  is  done  by  computing  what  is  known  as  the  prob- 
able error  of  the  mean  result.  The  computation  is  made  by 
writing  the  readings  in  a  vertical  column,  and  placing  opposite 
each  reading  the  difference  between  it  and  the  mean,  making  it 
plus  or  minus,  according  as  the  reading  is  greater  or  less  than 
the  mean.  This  difference  is  termed  the  "residual"  for  the  ob- 
servation in  question.  It  is  shown  in  the  theory  of  probabilities 
that  the  probable  error  of  the  mean  of  n  observations,  is 


=  ±  0.6745  *  T7ir-7T  (4) 


INTRODUCTION  7 

where  j  is  the  sum  of  the  squares  of  the  residuals  of  the  n  obser- 
vations. The  probable  error  is  not  the  most  probable  error,  nor 
is  it  a  limiting  value.  It  is  that  error  which  is  just  as  likely  to  be 
exceeded  as  not,  or  it  is  an  even  chance  that  the  error  is  smaller, 
as  that  it  is  larger  than  the  probable  error. 

In  addition  to  accidental  errors  of  observation,  there  are  to  be 
considered  the  errors  arising  from  faults  in  the  instrument  or  in 
the  method  of  observation.  These  are  classed  as  systematic 
errors  and  are  neither  to  be  eliminated  nor  estimated  by  computa- 
tion. They  cannot  be  removed  entirely,  but  may  be  minimized 
by  repeated  measurements  with  different  instruments  in  the  hands 
of  different  observers.  Much  thought  should  therefore  be  ex- 
pended upon  devising  correct  methods  of  observation,  and  means 
for  avoiding  systematic  errors,  since  upon  these  the  accuracy  of 
the  result  must  finally  depend.  Students  are  to  be  urged  to  use 
judgment  in  measurements  and  warned  against  excessive  care  in 
guarding  against  minute  mistakes,  while  errors  of  the  grossest 
kind  are  liable  to  be  made  in  the  process.  It  frequently  occurs 
that  a  student  in  measuring  the  length  of  a  wire  will  expend  much 
time  in  determining  the  length  to  hundredths  of  a  millimeter,  and 
yet  make  an  error  of  a  centimeter  in  the  result. 

7.  Influence  of  Errors  upon  the  Result.  It  is  frequently 
necessary  to  compute  a  result  from  one  or  more  quantities  ob- 
tained by  observation.  In  such  cases  it  is  of  interest  to  determine 
the  influence  of  errors  in  the  observed  quantities  upon  the  com- 
puted result.  If  A'  be  the  value  sought,  and  x  the  value  of  the 
quantity  observed,  then  X  is  some  function  of  x.  If  e  be  the 
error  in  x  due  to  all  causes,  and  £  the  error  in  X  consequent  upon 
e,  then 

/(*  +  *).  (5) 


It  may  be  shown  that  the  total  error  of  the  final  result  is  ap- 
proximately 


If  £±  be  the  error  in  the  derived  result  arising  from  an  error 
cl  made  in  the  determination  of  one  of  the  component  quantities, 


S  PHYSICAL    MEASUREMENTS 

and  £2,  -  -  £n,  be  the  errors  due  to  other  factors  entering 
into  the  final  result,  then  the  total  error  of  this  result  is,  in  the 
most  unfavorable  case,  B^  +  B.2  +  -  -  -  -En- 

It  is  very  instructive  to  calculate  roughly,  before  beginning  a 
given  experiment,  the  influence  which  a  small  error  in  the  evalua- 
tion of  each  factor  will  exert  upon  the  accuracy  of  the  final  result. 
Such  calculation  will  show  the  student  upon  which  steps  the 
greatest  care  should  be  expended,  and  also  at  what  points  an  at- 
tempt at  extreme  accuracy  means  simply  a  waste  of  time  and 
•energy. 

From  a  consideration  of  the  applications  of  the  above  formula 
valuable  suggestions  as  to  methods  of  measurement  are  often 
obtained,  whereby  the  percentage  of  error  may  be  much  reduced. 
Thus  let  it  be  required  to  determine  the  conditions  most  conducive 
to  accuracy  in  the  measurement  of  an  electrical  resistance  by  the 
slide  wire  bridge.  The  expression  for  the  resistance  measured 
by  means  of  a  slide  wire  bridge  is 

where  R  is  the  known  resistance  in  ohms,  a  the  reading  on  the 
bridge  wire,  c  scale  divisions  in  length.  In  this  case  the  expres- 
sion for  H  becomes 

E  =  e£~=eR          c__     8  (8) 

and  the  relative  error 

BE  c 

-X=^-=*^=a)'  (9) 

This  expression  will  be  a  minimum  for  a  maximum  value  of 
the  denominator,  a  (c  —  a).  But  since  a  -f-  (c  —  a)  =  c  is  a 
constant,  a  (c  —  a)  is  a  maximum  when 

a  =  c  —  a,  or  2  a  =  c  ; 

that  is,  the  adjustment  should  be  such  as  to  bring  the  reading  to 
the  middle  of  the  scale. 


INTRODUCTION  9» 

8.  Interpolation.  It  is  frequently  desirable  to  evaluate  a  phys- 
ical quantity  beyond  the  limit  of  the  subdivisions  of  the  instru- 
ment at  our  disposal.  Thus  let  it  be  required  to  weigh  a  body 
to  o.i  mg,  while  the  smallest  weight  in  the  box  of  weights  is  I 
mg;  or  let  it  be  required  to  determine  the  resistance  of  a  piece  of 
wire  to  o.i  ohm,  by  the  method  of  substitution  when  the  known 
resistance  is  subdivided  to  ohms  only.  In  such  cases  the  method 
of  interpolation  is  applied.  Thus  let  x0  be  the  observed  quantity 
corresponding  to  the  unknown  quantity  y0.  We  are  to  select  two 
quantities  X  and  x,  in  the  neighborhood  of  XQ  such  that  X  >  ^TO 
>  ,r.  Let  the  corresponding  values  of  y0  be  F  and  y.  Then  if 
these  values  lie  near  enough  together  so  that  we  can  assume  that 
F--V  is  proportional  to  X  —  ,r,  we  find 


9.     Hints  on  Computation.     The  following  suggestions  re- 
garding the  computation  of  results  will  be  found  useful. 

(a)  Taking  the  Mean.    In  taking  the  mean  of  a  set  of  read- 
ings, it  is  necessary  to  average  only  those  parts  of  the  various- 
readings  which  differ  among  themselves.    Thus  if  ten  readings 
have  the  first  three  figures  264,  and  differ  only  in  the  tenths  it 
is  clearly  unnecessary  to  average  more  than  the  tenths. 

(b)  Significant  Figures.    The  student  should  avoid  carrying, 
a  result  out  to  a  large  number  of  decimal  places,  far  beyond  the 
point  where  the  figures  have  any  significance  whatever.     Thus  if 
six  readings  of  the  barometer  be  743-3,  743-2,  743-3.  743-1*  743-2> 
743.3  mm.,  the  mean  is  743.233  mm.,  of  which  but  four  figures  are- 
significant  and  the  tenths  are  uncertain  since  they  cannot  be  read 
every  time  alike.    If  this  reading  be  corrected  for  temperature  the 
result  should  likewise  be  given  to  tenths  of  a  millimeter  but  no* 
farther,  since  nothing  is  known  beyond  that.     In  general  the  re- 
sult should  be  carried  to  so  many  figures  that  the  last,  owing  to 
errors,  makes  no  pretension  to  accuracy,  while  the  next  to  the  last 
may  be  regarded  as  reasonably  accurate. 

(c)  Approximations.    In  many  cases  where  it  is  necessary  to- 
compute  results  from  values,  some  of  which  are  very  small  in- 


IO  PHYSICAL    MEASUREMENTS 

comparison  to  others,  the  labor  may  be  greatly  reduced  by  approx- 
imation formulae.  Some  of  the  more  useful  are  given  below, 
where  a,  b,  c  and  d  are  to  be  regarded  as  very  small  in  compari- 
son to  unity. 

i±cm=i  ±tnd 


(II) 

=  i  n=  V*  d 


(i±a)   (i  ±&)  (i±c) 


(d)  Supplementary  Tables.  At  the  end  of  the  book  will  be 
found  a  series  of  tables  of  mathematical  and  physical  constants. 
The  student  will  find  it  of  advantage  to  consult  these  freely  in  the 
course  of  his  work.  While  the  values  of  the  physical  constants 
contained  in  these  tables  have  been  selected  from  reliable  sources, 
the  student  is  warned  against  the  error  of  assuming  that  a  value 
obtained  in  the  laboratory  is  necessarily  wrong,  because  it  differs 
slightly  from  that  given  in  the  table. 


CHAPTER  I. 

FUNDAMENTAL  MEASUREMENTS. 

10.  Fundamental    Magnitudes.      Most    physical    quantities 
may  be  expressed  either  directly  or  indirectly  in  terms  of  the 
fundamental  units  of  mass,  length,  and  time.    The  first  prob- 
lem of  physical  measurement  therefore  has  to  do  with  the  methods 
for  evaluating  certain  quantities  in  terms  of  these  fundamental 
magnitudes.     Closely  related  to  such  fundamental  measurements 
are  the  methods  employed  for  the  measurement  of  angles  and 
of  the  pressure  exerted  by  the  atmosphere.     On  account  of  the 
great  importance  of  these  quantities  the  methods  for  their  deter- 
mination are  included  here.     The  instruments  and  processes  de- 
scribed in  this  chapter  being  essentially  those  employed  in  the 
exercises  which  follow,  a  few  words  may  be  devoted  to  these  fun- 
damental measurements. 

LENGTH. 

11.  Contact  Meaurements.     Many  instruments  for  measure- 
ing  length  with  a  greater  or  less  degree  of  accuracy  involve  the 
necessity  of  bringing  the  instrument  into  contact  with  the  object 
to  be  measured.     This  may  be  effected  by  bringing  the  object 
between  two  jaws,  one  of  which  is  movable,  or  between  a  movable 
part  of  the  instrument  and  a  fixed  plane  of  reference.    In  any  case 
accuracy  of  measurement  requires  that  the  contact  between  the 
object  and  the  instrument  shall  be  exact,  and  that  the  distance 
between  the  points  of  contact  shall  represent  the  true  length  of  the 
object  to  be  measured. 

Hence  it  is  of  importance  to  see  that  the  contact  points  of  such 
instruments  are  scrupulously  clean  and  true,  and  that  the  ends  of 
the  body  are  accurately  faced  and  free  from  adhering  matter  of 
any  kind.  Such  measurements  are  termed  contact  measure- 
ments. 


12  PHYSICAL    MEASUREMENTS 

12.  Methods  of  Subdivision.  Instruments  for  refined  meas- 
urements of  length  usually  involve  the  principle  of  the  microm- 
eter screw  or  of  the  vernier,  or  a  combination  of  the  two.  Prom- 
inent among  such  instruments  may  be  mentioned  the  micrometer 
gauge,  the  spherometer,  the  vernier  caliper,  the  cathetometer, 
the  micrometer  cathetometer,  the  comparator,  and  the  dividing 
engine.  Of  these  the  micrometer  gauge  and  the  spherometer 
employ  the  principle  of  the  micrometer  screw,  the  vernier  caliper 
and  the  ordinary  cathetometer  employ  the  principle  of  the  vernier, 
while  the  micrometer  cathetometer,  the  comparator  and  the  divid- 
ing engine  employ  a  combination  of  the  two.  It  is  not  the  inten- 
tion to  describe  the  working  of  each  of  these  instruments  in  de- 
tail, but  so  to  give  fundamental  principles  upon  which  each 
instrument  is  based,  as  to  enable  the  student  to  make  the  applica- 
tion for  himself.  *' 

13.  The  Micrometer  Screw.     In  the  micrometer  screw  we 
have  a  screw  of  fine  thread  working  in  a  nut  and  furnished  with  a 
graduated  head  divided  into  some  convenient  number  of  aliquot 
parts.     A  complete  rotation  of  the  head  advances  or  withdraws 
the  screw  by  an  amount  equal  to  the  distance  between  its  adjacent 
threads,  that  is  by  some  fraction  of  a  centimeter,  or  of  an  inch. 
This  small  length  is  further  subdivided  by  means  of  the  divisions 
on  the  graduated  head,  so  that  the  least  .count  of  the  instrument, 
that  is,  the  least  length  directly  measurable  by  it,  is  the  distance 
between  the  threads,  divided  by  the  number  of  divisions  on  the 
head.     Still  finer  readings  may  be  made  by  estimating  tenths  of 
these  divisions. 

14.  Exercise  i.     The  Micrometer  Gauge.     In  the  microm- 
eter gauge   the  end  of  the  screw  works   against   a  fixed  jaw. 

.The  number  of  complete  turns  of 
the  screw  is  read  from  the  stem 
of  the  instrument  and  the  frac- 
tion of  a  turn  from  the  gradu- 
ated head.  In  use  the  least  count 
of  the  instrument  is  first  deter- 
mined and  recorded.  The  end  of 
-  *•  the  screw  is  next  brought  into 


FUNDAMENTAL   MEASUREMENTS 


contact  with  the  fixed  jaw  by  slight  pressure  and  the  zero  reading 
taken.  The  object  to  be  measured  is  then  brought  between  the 
jaws  and  the  screw  turned  down  until  contact  is  made  as  before 
and  the  reading  is  again  taken.  The  difference  between  this  read- 
ing and  the  zero  reading  gives  the  thickness  of  the  object,  ex- 
pressed in  the  units  marked  upon  the  stem.  In  determining 
the  zero  and  final  readings  the  mean  of  five  readings  is  to  be  taken 
in  each  case.  In  more  accurate  instruments  undue  pressure  upon 
the  jaws  is  avoided  by  means  of  a  ratchet  head  which  slips  as  soon 
as  contact  is  made.  In  using  such  instruments  always  turn 
slowly  by  means  of  this  head  and  stop  as  soon  as  the  ratchet 
slips. 

FORM    OF   RECORD. 

Exercise  i.     The  Micrometer  Gauge. 

.  Date 

Object    measured Pitch  of  screw 

Micrometer   gauge    No Least    count 

Zero    readings  Final   readings 


Mean    Mean    

Thickness 

15.     Exercise  2.     The  Spherometer.       In  the  spherometer 

the  screw  works  in  a  nut  supported  by  a  frame  having  three  legs 

of  equal  length,  so  placed  that  when 
the  four  points  rest  upon  a  plane  the 
three  feet  form  an  equilateral  triangle 
about  the  point  of  the  screw  at  the 
center.  The  instrument  is  usually 
placed  on  a  square  of  good  plate  glass. 
Notice  of  contact  between  the  point  of 
the  screw  and  the  plane  is  given  by  the 
instrument's  hobbling  or  rocking  on 
the  plane.  The  screw  is  then  care- 
fully turned  back  until  this  hobbling 
just  ceases.  The  zero  reading  is  then 
taken.  The  object  to  be  measured  is  then  placed  beneath  the  mid- 
dle point  and  the  screw  turned  down  until  contact  is  made,  and  the 


Fig.  2. 


PHYSICAL    MEASUREMENTS 


reading  taken  as  before.     The  difference  between  the  zero  and 
final  readings  gives  the  thickness  of  the  object. 

In  the  more  delicate  form  of  the  instrument,  made  by  the  Ge- 
neva Society,  (Fig.  3),  the  screw  point  is  connected  by  a  system 
of  light  levers,  to  a  delicate  pointer  which  rises  when  contact  is 
made.  Readings  are  taken  when  the  pointer  rises  to  a  fixed  mark. 
In  use  avoid  touching  the  graduated  head  with  the  fingers.  Turn 
by  means  of  the  milled  head  provided  for  that  purpose. 

An  extremely  d  >e  1  i- 
cate  means  of  determin- 
ing the  position  of  con- 
tact in  the  use  of  the 
spherometer,  is  by 
means  of  the  interfer- 
ence fringes  of  sodium 
light.  The  spherometer 
is  placed  upon  a  piece  of 
good  plate  glass  and  a 
J  small  piece  of  glass  with 
B  a  good  plane  surface  is 
placed  under  the  central 
leg.  When  the  surface 
of  the  glass  is  lighted  by 
a  sodium  flame,  the  in- 
terference fringes  appear  at  once.  The  slightest  increase  in  pres- 
sure causes  the  lines  to  shift  their  position,  thus  indicating  the 
position  of  contact. 

FORM    OF   RECORD. 

Exercise  2.     The  Spherometer.     To  measure  the  thickness  of 
a  piece  of  glass. 

Pitch    of    screw 

Spherometer    No 

Object  measured 

Zero   readings  '  Fi'naY  readings 


Date 
Least 


Count. 


Mean 


Thickness  of 


Mean 


01    23456789   10  1 

I     I      I     I     I      I     I      I     I      I     I      I 


FUNDAMENTAL,   MEASUREMENTS  15 

1 6.  The  Vernier.  The  vernier  is  a  device  for  subdividing 

the  least  division  of  a  scale. 

_!£  It  consists  of  a  short  subsidi- 
ary scale  placed  in  front  of, 
Flg*  4*  and  in  contact  with  the  scale 

of  the  instrument,  and  is  usually  so  divided  that  n  divisions  of  the 

vernier     correspond     to 

n  — •  i    divisions   of   the 

scale.     The  least  count, 

i.  e.,  the  least  subdivision 

which   may   be   read   by 

the  use  of  the  vernier,  is 

i/n  of  a  scale  division. 

Thus,  if  S  be  the  least 

division  of  the  scale  and 

V  the  least  division  of  the  \ernier  then 


Fig.  5. 


V  — 


or 


Least  count:  S—  V  = 


(12) 


In  some  cases  n  divisions  on  the  vernier  are  made  equal  to  one 
less  than  some  multiple  of  n  divisions  of  the  scale;  the  formula 
then  becomes 

nV—(.an—  i)  S 
whence 

Least  count:  aS—  V—  — 5*  (13) 

as  before. 

To  read  the  vernier,  first  read  the  units  of  the  scale  up  to  the 
zero  of  the  vernier;  to  this  reading  add  as  many  nths  of  a  scale 
division  as  are  indicated  by  the  vernier  division  which  coincides 
with  a  scale  division.  Thus  in  Fig.  4  the  reading  is  2.6  scale 
divisions. 

17.  Exercise  3.  The  Vernier.  Determine  the  least  count 
of  the  verniers  on  five  or  more  instruments. 


i6 


PHYSICAL,    MEASUREMENTS 


FORM   OF  RECORD 

Exercise  3.     To  determine  the  least  count  of  the  verniers  on 
five  different  instruments  assigned  by  the  instructor. 


Date. 


Name  of  Instrument 


Value  of  S 


1 8.  Exercise  4.  The  Vernier  Caliper.  The  vernier  caliper 
(Fig.  6),  is  an  instrument  in  which  the  principle  of  the  vernier  is 
applied  to  the  measurement  of  length.  It  consists  of  a  pair  of 
steel  jaws,  one  of  which  is  attached  to  the  scale,  the  other  to  the 

vernier  which  slides 
upon  the  scale.  In 
some  instruments  the 
movable  jaw  is  pro- 
ind  slow  motion  screw 
for  fine-  adjustment. 

Fig.  6.  '^Q-^        Most    ins  truments 

are  adapted  to  inside  measurements  also,  by  means  of  a  pair  of 
rounded  lugs  attached  to  the  ends  of  the  jaws.  When  a  separate 
scale  is  not  provided  for  inside  measurements,  the  thickness  of 
these  lugs  must  be  added  to  the  indicated  reading. 

In  use  the  value  of  a  scale  division  S,  and  the  least  count,  are 
first  determined  and  recorded.  The  jaws  are  next  brought  to- 
gether and  the  zero  reading  taken.  The  object  to  be  measured  is 
then  placed  between  the  jaws  and  the  mean  of  several  readings 
taken.  The  difference  between  the  zero  and  final  readings  gives 
the  length  of  the  object. 

FORM   OF  RECORD 

Exercise  4.  '  The  Vernier  Caliper.  To  measure  the  diameter 
and  length  of  a  brass  cylinder. 


Object    measured.... 
Vernier    caliper    No. 
Zero  readings 


Date    

Least   count    

Diameter   readings  Length   readings 


Mean 


'Mean 


Diameter 


Mean 


Length 


FUNDAMENTAL  MEASUREMENTS  17 

19.  Line  Measurements.     In  the  previous  exercises  on  meas- 
urements of  length  it  will  be  noted  that  in  each  case  the  meas- 
urement has  been  effected  by  contact  measurements,  that  is,  by 
bringing  the  measuring  instruments  into  contact  with  the  object  to 
be  measured.     Frequently  this  is  neither  practicable  nor  desirable. 
In   such   cases   line   measurements   are   employed ;   that   is,   the 
distance  between  two  lines  drawn  upon  a  body  is  determined, 
by  focusing  a  microscope  or  telescope  upon  the  lines  in  succes- 
sion, and  noting  the  readings  of  the  vernier  attached  to  the  instru- 
ment.   Line  measurements  are  made  by  means  of  the  cathetom- 
eter, the  comparator  and  the  dividing  engine,  and  are  used  in  all 
cases  where  great  accuracy  is  desired. 

20.  The    Cathetometer.     The    cathetometer    is    an    instru- 
ment for  measuring  the  difference  in  height  between  two  points. 
It  consists  of  a  vertical  standard,  (Fig.  7),  upon  which  slides  a 
ring  carrying  a  telescope  and  furnished  with 

a  vernier  and  a  clamp  for  holding  the  tele- 
scope at  any  height  desired.  The  instrument 
is  provided  with  a  level  for  bringing  the  axis 
of  the  telescope  into  a  horizontal  line,  by 
means  of  adjusting  screws.  In  most  instru- 
ments the  level  may  be  rotated  through  a 
right  angle  about  the  standard  as  an  axis,  in 
order  to  secure  the  vertical  position  of  the 
latter.  The  telescope  is  focused  first  upon 
one  point  and  the  reading  taken,  and  then 
lowered  by  means  of  the  clamp  to  the  level  ^ 
of  the  second  point,  and  set  upon  it,  and  Fig.  7. 

the  corresponding  reading  determined.  For  ease  and  accuracy 
in  making  the  settings,  the  ring  carrying  the  telescope  is  usually 
furnished  with  a  slow  motion  screw.  The  difference  between  the 
two  settings  gives  the  vertical  distance  between  the  two  points. 

21.  Optical   Micrometers.     A   micrometer   is   a   device   for 
measuring  minute  lengths  or  changes  in  length  with  great  accu- 
racy.     Such    instruments    are   constantly   employed    in    physical 
measurements  and  are  made  in  a  variety  of  forms.     Since  most 


jg  PHYSICAL    MEASUREMENTS 

such  instruments  embody  a  combination  of  a  micrometer  screw 
and  an  optical  system  of  some  sort,  it  has  seemed  fitting  to  class 
them  as  optical  micrometers. 

(a)  Eye-piece  micrometer.  Perhaps  the  simplest  form  of  such 

micrometer  consists  of  a  finely  divided  scale, 
ruled  upon  glass  and  placed  upon  a  diaphragm 
between  the  two  lenses  in  the  eye-piece  of  a  mi- 
croscope. The  diaphragm  must  be  so  adjusted 
that  the  scale  divisions  are  seen  sharply  defined 
on  looking  through  the  eye-piece  turned  toward  a 
bright  window.  On  introducing  the  eye-piece 
again  into  the  microscope  and  focusing  it  upon 
some  minute  object,  the  image  will  be  seen  sharply 
outlined  upon  the  ruled  surface  and  the  size  of  the 
image  may  be  read  off  directly  in  terms  of  the  units  of  the  scale. 
By  placing  upon  the  stage  of  the  microscope  a  test  plate,  or 
object  micrometer,  containing  divisions  of  a  known  length,  the 
magnifying  power  of  the  instrument  and  the  corresponding 
value  of  a  single  division  of  the  eye-piece  scale  may  be  directly 
determined.  Once  the  value  of  a  division  is  known  the  eye-piece 
may  be  used  to  measure  lengths  by  simply  counting  the  number  of 
scale  divisions  covered  by  the  magnified  image. 

(b)  Screzv  micrometer.     In  the  screw  micrometer  (Fig.  9), 
the  tube  of  the  microscope  is  moved  in  a  line  at  right  angles  to 
its  axis,  by  means  of  a  micrometer  screw,  and  the   different 
parts  of  the  image  may  be  brought  successively  upon  the  cross 
hair  of  the  microscope.     The  advantage  of  such  an  instrument 
is  that  its  range  of  measurement  may  be  extended  to  several 
centimeters. 

When  fitted  to  a  microscope  of  low  power  and  mounted  upon 
a  vertical  standard  the  instrument  forms  a  micrometer  catheto- 
meter  (Fig.  10),  and  may  be  used  to  determine  the  distance 
between  adjacent  points  with  great  accuracy.  It  may  also  be 
used  to  measure  small  changes  in  length  as  in  the  determination 
of  Young's  modulus  by  stretching,  (Exercise  18). 

(c)  Filar  micrometer.     In  the  filar  micrometer   (Fig.   n), 


FUNDAMENTAL   MEASUREMENTS  IQ 

the  micrometer  screw  is  used  to  displace  across  the  field  of  view 
a  glass  plate  carrying  a  pair  of  fine  lines,  or  a  light  frame  upon 
which  are  stretched  a  pair  of  spider  threads,  either  parallel  and 
close  together  or  crossing  at  an  acute  angle.  In  most  instruments 
there  are  also  two  fixed  threads  or  hairs,  at  right  angles  to  one 


Fig.  9. 


Fig.  10. 


another.  All  the  threads  or  cross  hairs  must  lie  in  the  focal 
plane  of  the  eye-lens  and  be  seen  sharply  defined  when  the  micro- 
meter is  turned  toward  the  bright  sky. 

When  the  micrometer  is  placed  in  the  microscope  or  telescope 
and  the  instrument  focused  upon  some  object,  the  measurement 
of  any  featare  of  the  image  is  effected  by  bringing  the  movable 
hairs  successively  into  coincidence  with  the  points  under  con- 


20  PHYSICAL    MEASUREMENTS 

sidcration  and  noting  the  corresponding  readings  upon  the 
micrometer  head.  In  some  instruments  the  count  for  the  integral 
number  of  turns  of  the  screw  is  shown  in  the  field  of  view  by 
means  of  a  toothed  index,  where  the  moving  hairs  pass  from  one 
tooth  to  the  next  for  one  complete  revolution  of  the  head.  Every 
fifth  notch  is  made  deeper  than  the  others  to  assist  in  the  reading. 
The  zero  may  be  taken  at  any  definite  point  in  the  field  as  best 
suits  the  convenience  of  the  observer.  Usually  the  divided  head 
is  held  in  place  by  means  of  a  friction  washer  or  a  lock  nut  which 


Fig.  ii. 

allows  it  to  be  adjusted  for  any  desired  zero.  If  there  be  a 
fiducial  mark  in  the  field  of  view  to  be  used  as  a  zero,  then  when 
the  parallel  hairs  include  this  mark  symmetrically  between 
them  the  divided  head  should  read  zero,  or  be  adjusted  until  it 
does. 

The  micrometer  here  described  is  used  in  the  micrometer  micro- 
scopes on  dividing  engines  and  comparators,  in  the  reading  micro- 
scopes of  spectrometer  and  other  finely  divided  circles,  in  tele- 
scopes on  cathetometers  and  spectrometers,  and  on  astronomical 
telescopes.  The  filar  micrometer  is  also  used  for  measurement  of 
small  angles,  as  shown  in  Article  33. 

It  is  to  be  remembered  that  lengths  measured  by  a  micrometer 
placed  in  the  eye-piece  of  a  microscope  are  made  upon  the  magni- 
fied image  and  are  therefore  related  directly  to  the  magnifying 
power  of  the  instrument.  Consequently  in  any  series  of  measure- 
ments this  magnifying  power  must  be  kept  constant,  i.  e.t  neither 


FUNDAMENTAL    MEASUREMENTS  21 

the  microscope  objective  nor  the  distance  between  the  objective 
and  the  micrometer  must  be  changed. 

It  is  also  to  be  observed  that  in  making  any  setting  the  hairs 
must  be  brought  up  to  the  final  position  from  the  same  direction 
in  each  case,  in  order  to  avoid  the  lost  motion  of  the  screw.  If 
this  position  has  been  passed,  the  screw  should  be  run  back 
through  at  least  two  turns  and  brought  up  with  greater  care. 
Usually  the  hairs  are  carried  from  left  to  right  across  the  field  by 
direct  pressure  of  the  screw,  and  displaced  in  the  opposite  direc- 
tion by  means  of  a  spring  as  released  by  the  screw.  Hence  the 
best  direction  for  approach  to  a  setting  is  that  in  which  the  spring 
is  compressed. 

22.  The  Dividing  Engine.  In  the  dividing  engine  the 
micrometer  screw  is  mounted  in  a  rigid  metallic  bed  so  as  to  rotate 
freely,  and  carries  upon  it  the  movable  nut  which  advances  or 
recedes  as  the  screw  rotates.  Attached  to  this  nut  is  the  traveling 
carriage  upon  which  is  mounted  the  device  for  ruling  short  lines 
transverse  to  the  length  of  the  screw.  This  ruling  device  may  be 
operated  either  by  hand  or  automatically  by  the  machine  itself, 
and  is  so  arranged  as  to  make  every  fifth  and  tenth  line  dis- 
tinctively longer  than  the  others,  thereby  facilitating  the  reading 
of  the  graduated  divisions. 

In  modern  machines  the  pitch  of  the  screw  is  usually  one  milli- 
meter and  the  divided  head  allows  the  screw  to  be  advanced 
through  any  desired  fraction  of  this  length.  Usually  the  smallest 
division  of  the  head  may  be  still  further^  subdivided  by  a  small 
vernier.  In  most  machines  the  carriage  is  provided  with  one  or 
more  micrometer  microscopes,  whereby  the  operator  is  enabled  to 
extend  the  graduation  to  lengths  beyond  the  length  of  the  screw. 

In  the  machine  shown  in  Fig.  12,  the  graduating  device  and  the 
reading  microscopes  are  fixed  and  the  table  carrying  the  surface  to 
be  graduated  or  the  scale  to  be  examined,  moves  with  the  motion 
of  the  screw.  Otherwise  the  relations  are  the  same  as  those  de- 
scribed. 

The  dividing  engine  may  also  be  used  to  measure  lengths 
smaller  than  that  of  the  screw  with  great  accuracy,  as  soon  as  the 
pitch  of  the  screw  is  known.  To  determine  this,  a  standard  scale 


22  PHYSICAL    MEASUREMENTS 

is  supported  horizontally  beneath  the  reading  microscope,  and 
accurately  parallel  to  the  length  of  the  screw.  This  adjustment 
is  secured  when  the  scale  is  seen  in  sharp  focus  throughout  its  en- 
tire length  as  the  microscope  is  moved  over  it,  and  when  the  inter- 
section of  the  cross  hairs  in  the  microscope  cuts  the  divisions  of 


Fig.  12. 

the  scale  at  corresponding  points  throughout.  The  microscope  is 
then  set  upon  some  definite  division  of  the  standard  scale  and  the 
number  of  turns  of  the  screw  needed  to  carry  the  microscope  to 
the  next  similar  division  is  carefully  determined.  From  the 
value  of  the  scale  division  measured  and  the  observed  rotation, 
the  pitch  of  the  screw  is  readily  computed. 

The  dividing  engine  is  also  useful  for  measuring  accurately 
small  lengths,  for  calibrating  thermometer  or  other  capillary 
tubes,  for  study  of  the  errors  of  standard  scales,  for  de- 
termining the  constants  and  errors  in  the  ruling  of  diffraction 
gratings,  etc. 

23.  Exercise  5.  The  Dividing  Engine.  In  order  to  meas- 
ure with  the  dividing  engine  the  distance  between  any  two  fixed 
points,  the  line  connecting  the  points  in  question  is  placed  accu- 
rately parallel  to  the  screw  of  the  instrument  and  readings  taken 
first  upon  one  point  and  then  upon  the  other.  The  difference 
between  the  means  of  the  two  sets  of  readings  gives  the  distance 
sought.  In  making  any  setting  of  the  microscope  care  must  of 
course  be  taken  to  avoid  lost  motion  of  the  screw. 


FUNDAMENTAL  MEASUREMENTS  23 

A  smoothly  planed,  rectangular  iron  bar  has  ruled  upon  one  of 
its  sides  two  fine  lines  at  right  angles  to  its  length.  A  third  fine 
line  ruled  lengthwise  of  the  bar,  intersects  the  transverse  lines 
normally,  and  enables  the  operator  to  set  the  bar  accurately  parallel 
to  the  length  of  the  screw.  The  temperature  of  the  bar  is  noted 
and  at  least  five  readings  made  upon  each  of  the  transverse  marks. 
The  observed  distance  is  to  be  corrected  for  temperature  by  means 
of  Table  X. 

FORM   OF  RECORD 

Exercise  5.  The  Dividing  Engine.  To  determine  the  distance 
between  two  marks  on  an  iron  bar. 


Date 

Temperature Reading  A.          Reading  B.          Difference 


Distance  corrected  to  2O°C. 


ANGLE. 


24.  Measurement  of  Angle.     One  of  the  most  common  meas- 
urements to  be  made  in  the  .physical  laboratory  is  the  determination 
of  an  angle.     This  may  be  the  angle  included  between  two  lines 
on  a  plane  surface,  or  that  subtended  by  two  points  in  space,  or 
the  angle  swept  out  by  a  body  as  it  rotates  about  an  axis.     For 
the  determination  of  angular  magnitudes  many  devices  are  em- 
ployed of  which  only  the  more  important  will  be  described  here. 

25.  Definitions.     In  circular  measure  an  angle  0  is  defined 
as  the  ratio  between  its  subtending  arc  s,  and  the  radius  r,  or 

e=    -£-.  (14) 

The  unit  angle  in  circular  measure  is  the  radian,  that  is,  an 
angle  whose  subtending  arc  is  equal  to  the  radius.  Angles  are 
also  measured  in  degrees,  minutes,  and  seconds  (°  '  ").  The  de- 
gree is  the  angle  subtended  by  1/360  part  of  a  circumference. 


PHYSICAL  MEASUREMENTS 


From  this  it  follows  that  the  entire  angle  that  may  be  described 
about  a  point  is  2  TT  radians,  or  360°,  and  consequently 

i  radian  =  ~~  =  57°-295& 

7T 

i  degree  —  radians. 

In  theoretical  formulae  angles  are  usually  expressed  in  radians, 
while  measuring  instruments  give  the  values  in  degrees  and  frac- 
tions of  a  degree,  hence  frequent  use  must  be  made  of  the  above 
equivalents  in  reducing  from  one  system  to  the  other. 

Most  measuring  instruments  are  graduated  in  degrees  and 
minutes  and  furnished  with  verniers  or  reading  microscopes  for 
subdividing  the  least  division  of  the  scale.  The  degree  of  accuracy 
required  in  the  specific  problem  in  hand  must  determine  the  re- 
finements in  measurement  to  be  employed. 

26.     Exercise  6.     The  Protractor.     In  determining  the  angle 

included  between  two  lines 
drawn  upon  a  plane  surface, 
as  a  sheet  of  paper  of  a  black- 
board, one  of  the  simplest 
methods  is  to  use  a  protractor 
(Fig.  13).  This  is  a  gradu- 
ated circle  with  an  arm  rotat- 
ing about  its  center.  The  cen- 
ter of  the  instrument  is  placed 
over  the  point  of  intersection 
of  the  two  lines  in  question 
and  the  edge  of  the  movable 
arm  is  brought  into  coinci- 
dence with  the  lines  in  turn 
and  the  corresponding  read- 
ings of  scale  and  vernier  noted 
in  each  case.  The  difference 
between  these  readings  is  the 
angle  sought.  Measure  the 


Fig.  13. 


the  angles  of  a  triangle  and  check  work  by  taking  the  sum.     Meas- 
ure each  angle  three  times. 


FUNDAMENTAL  MEASUREMENTS 


FORM  OF  RECORD 

Exercise  6.     The  Protractor.     To   measure   the   angles  of  a 
triangle. 

Date. . 


Angle 


Mean 


B 


•Sum  =: 


27.     Exercise  7.     Angles  from  Trigonometric  Functions.     A 

very  simple  method  for  the  approximate  determination  of  an 
angle  is  by  means  of  measurements  which  enable  us  to  compute 
either  its  sine  or  tangent,  preferably  the  latter.  Thus  if  it  be 
desired  to  know  the  angle  <f>  subtended  at  the  eye  or- at  the  ob- 
jective of  a  telescope  by  a  length  /,  placed  symmetrically  normal 
to  the  line  of  sight,  at  a  distance  L,  a  simple  consideration  shows 
that  the  angle  is  given  by 


tan =  — — 

2  2.L, 


d5) 


from  which  </>  may  be  readily  evaluated. 

Determine,  both  in  radians  and  in  degrees,  the  angle  subtended 
by  some  object  in  the  laboratory  at  three  different  distances,  and 
compare  results  thus  obtained.  To  what  degree  of  approximation 
may  angles  of  i°,  2°,  3°,  5°,  and  10°  be  set  equal  to  their  tan- 
gents. 

FORM   OF  RECORD 

Exercise  J.  To  determine  the  angle  subtended  by  an  object  at 
three  different  distances. 


Object 

0 

Da 

tan 

te                    

I 

L 

tan  0/2 

0 

Difference 

i 

2 

3 

0  in  degrees 
0  in  radians 

i° 

2° 

3° 

5° 

10° 

tan   <f> 
Difference 

PHYSICAL    MEASUREMENTS 


28.  Correction  for  Eccentricity.  Instruments  intended  for 
accurate  work  are  provided  with  two  or  more  verniers  for  the 
purpose  of  eliminating  the  error  in  reading  the  graduated  circle 

iue  to  eccentricity.  This 
;rror  is  due  to  the  fact  that  the 
renter  of  the  graduated  circle 
and  the  center  about  which  it 
or  the  vernier  arms  rotate, 
are  never  exactly  coincident. 
This  error  is  usually  small, 
yet  for  instruments  of  preci- 
sion it  must  be  eliminated. 
This  is  readily  accomplished 
by  reading  "the  circle  at  two 
points,  180°  apart  and  combin- 
ing the  readings  for  every 
observation. 

Thus  in  Fig.  14,  let  C  be  the  true  center  of  the  graduated  circle, 
and  let  C'  be  the  center  of  rotation  either  for  the  circle  or  for  the 
arm  carrying  the  vernier.  Then  when  the  vernier  stands  at  A, 
the  axis  of  the  telescope  points  in  the  direction  of  CA,  while  the 
reading  at  A  corresponds  to  the  direction  C'A,  and  the  error  e, 
due  to  the  displaced  center  is  the  angle  CAC  or  A'C'A.  The  true 
reading  is  the  arc  ZA'  instead  of  ZA.  On  the  other  hand,  the 
reading  on  vernier  B,  corresponds  to  a  direction  C'B,  while  the 
telescope  looks  in  the  direction  CB,  and  the  error  e,  is  BC'B'  as 
before.  In  the  first  case  the  true  reading  R,  is  given  by 


(16) 


and  in  the  second  by 
whence 


R  = 


R  + 180°  —B  —  e 
A+  (£  —  180° 


(17) 


the   error   due   to   eccentricity   may   be   completely   eliminated 
by  taking  the  mean  of  two  readings  180°  apart.     In  practice  it 


777  ,  777 


FUNDAMENTAL   MEASUREMENTS  2J 

is  well  to  designate  the  right  hand  vernier  as  A  and  record  the  de- 
grees as  read  from  this  vernier  only,  averaging  the  minutes  and 
seconds  as  read  from  the  two.  It  may  also  be  shown  that  the 
error  due  to  eccentricity  may  be  eliminated  by  taking  the  mean 
reading  of  any  number  of  equidistant  verniers. 

29.  Telescope  and 
Scale.  In  measuring 
small  anguilar  displace- 
ments as  indicated  by 
the  tilting  or  rotating 
of  a  mirror  about  an 
axis  the  most  common 
and  efficient  method  is 
that  of  the  telescope 
and  scale.  A  telescope 
of  low  magnifying  power  (Fig.  15)  is  furnished  with  a  scale, 
usually  in  millimeters,  on  which  the  figures  are  both  inverted 
and  perverted.  This  scale  is  so  arranged  that  it  may  be  held 
near  the  telescope  and  at  right  angles  to  its  optical  axis,  in 
either  a  horizontal  or  vertical  position.  A  small  plane  mirror  m, 
whose  angular  displacement  is  to  be  determined,  forms  a  virtual 
image  of  the  scale  which  is  viewed  by  the  telescope.  The  scale 
is  made  as  nearly  parallel  to  the  surface  of  the  mirror  as  may 
be  and  the  telescope  adjusted  until  the  scale  division  nearest  the 
axis  of  the  instrument  is  seen  sharply  defined  on  the  cross  hairs. 

If  now  the  mirror  and  its  normal  be  rotated  through  an  angle 
0,  then  by  the  law  for  .the  reflection  of  light,  the  reading  r,  on  the 
scale  as  seen  in  the  telescope  will  form  an  angle  of  2  9  with  the 
axis  of  the  instrument.  If  D  be  the  distance  from  the  scale  to  the 
mirror  then 


Fig.  15. 


tan  2  0  — 


D 


(18) 


from  which  the  value  of  6  is  readily  determined.     For  accurate 
measurements  the  deflections  are  read  first  upon  one  side  and 


28 


PHYSICAL    MEASUREMENTS 


then  upon  the  other,  and  the  mean  value  for  r  used  in  computa- 
tion. The  equation  (18)  also  shows  that  the  sensitiveness  of  the 
method  increases  as  the  distance  D  is  increased. 

In  many  cases  the  ratio  between  the  deflections  is  all  that  is 
needed.  In  such  cases  if  two  scale  readings  rt  and  r2  have  been 
observed  we  have 


tan 


tan 


(19) 


or,  since  for  small  angles  the  tangents  may  be  set  equal  to  the 
angles  themselves,  we  see  that  the  angular  deflections  are  directly 
proportional  to  the  scale  readings. 

The  method  of  telescope  and  scale  is  of  universal  application  in 
work  with  galvanometers,  magnetometers,  radiometers  and  in 
all  cases  where  slight  angular  displacements  are  to  be  determined. 
An  exceedingly  delicate  instrument  embodying  the  same  principle 
is  found  in  the  optical  lever. 

30.     Exercise  8.     The  Optical  Lever.     The  optical  lever  is  a 

device  for  measuring 
small  lengths  by  means 
of  the  displacements  of  a 
beam  of  light  reflected 
from  a  plane  mirror.  It 
consists  of  a  stout  bar 
(Fig.  16),  supported  up- 
on four  short,  blunt 
pointed  legs,  of  which 
one,  at  one  end,  is  a  mi- 


Fig.  16. 


crometer  screw,  for  pur- 
poses of  adjustment  and  calibration.  The  bar  carries  at  its 
middle  point  a  plane  mirror  capable  of  rotation  about  a  horizontal 
axis,  and  which  may  be  clamped  in  any  desired  position. 

The  instrument  is  placed  upon  a  piece  of  good  plate  glass,  in 
front  of  and  some  four  or  five  meters  distant  from  a  reading  tele- 
scope furnished  with  vertical  scale.  The  telescope  is  focused 
upon  the  reflected  image  of  the  scale  as  seen  in  the  mirror,  and  the 


FUNDAMENTAL   MEASUREMENTS  29 

micrometer  screw  adjusted  till  all  hobbling  ceases.  The  mirror 
is  then  rotated  until  the  central  division  of  the  scale  falls  upon  the 
horizontal  cross  hair  in  the  telescope.  In  making  these  adjust^ 
ments  care  must  be  taken  to  free  the  plate  and  the  points  of  the 
feet  from  all  dust  particles  or  lint,  as  otherwise  the  zero  point 
will  vary  after  the  instrument  has  been  tilted  or  moved  on  the 
plane. 

After  these  adjustments  have  been  made  the  object,  as  a  small 
piece  of  microscope  slide,  is  placed  under  the  middle  of  the  lever, 
the  instrument  is  tilted  so  as  to  stand  upon  three  legs,  and  the 
reading  on  the  scale  noted.  The  lever  is  then  tilted  in  the  oppo- 
site direction,  placing  a  small  weight  if  necessary  upon  the  end 
previously  elevated,  and  the  scale  reading  again  noted.  The  two 
positions  of  the  lever  are  shown  in  Fig.  17.  Then  if  the  scale 


m 


Fig.  17. 


readings  in  the  two  tilted  positions  of  the  mirror  be  r±  and  rs,  if 
L  be  the  distance  from  scale  to  mirror,  /  the  half  length  of  the 
lever  and  a  the  angle  of  tip  in  each  case,  we  have  for  d,  the 
thickness  of  the  glass  to  be  measured 


d  =  /sin  a  =  la 


20 


since  a  is  small. 


PHYSICAL    MEASUREMENTS 


Also  since  the  total  angular  displacement  of  the  mirror  is  2  a, 
the  reflected  beam  is  turned  through  4  a,  and  consequently 


=  2  tan  2  a  =  4  tan  a 


and 


d  = 


(21) 
(22) 

(23) 


Make  at  least  three  determinations  of  r±  and  r2  and  take  the 
mean.     Determine  /  by  means  of  the  dividing  engine. 


FORM   OF  RECORD 


H.vtfcise  8.     The  Optical  Lever.     To  measure  the  thickness  of 
a  piece  of  glass. 

Date 

Object  measured 


I 

Mean 


Y\  —  Tz  ^^ 


d  — 


31.  The  Level  Tester.  The  spirit  level  is  an  instrument  for 
testing  a  line  or  a  surface  as  to  its  deviation  from  the  true 
horizontal.  It  consists  of  a  tube  (Fig.  18),  nearly  rilled  with 


Fig.  18. 

ether  or  alcohol  and  closed  at  both  ends.  The  inner  surface  of 
the  upper  side  of  the  tube  is  accurately  ground  into  a  circular  arc 
of  large  radius.  When  the  tube  is  brought  into  the  true  horizontal 


FUNDAMENTAL   MEASUREMENTS  3! 

the  enclosed  air  bubble,  seeking  the  highest  point  of  the  arc, 
stands  at  the  middle  of  the  tube.  Any  change  from  the  horizontal 
is  indicated  by  the  bubble  moving  toward  the  higher  end.  A  scale 
of  equal  divisions  is  graduated  upon  the  tube  for  reading  the 
position  of  the  bubble. 

In  accurate  instruments  the  tube  is  held  in  its  frame  by  adjust- 
ing screws  so  that  its  ends  may  be  raised  or  lowered  to  bring  the 
bubble  into  the  middle  when  the  level  is  placed  upon  a  horizontal 
plane.  By  the  horizontal  screws  the  tube  may  be  shifted  laterally 
so  as  to  bring  its  axis  parallel  to  the  line  whose  inclination  is  to 
be  tested.  In  sensitive  levels  the  vertical  adjustment  is  seldom 
perfect,  but  it  is  more  convenient  to  eliminate  this  error  in  use 
than  to  correct  it  by  adjustment.  This  is  done  by  reading  the 
level  in  the  direct  and  reversed  positions  each  time  and  taking 
half  the  difference  between  the  readings  as  the  true  position  of 
the  bubble. 

32.  Exercise  9.  Constants  of  the  Level.  The  value  of 
one  division  of  the  level  and  the  radius  of  curvature  of  the 
tube  are  termed  the  constants  of  the  level.  These  may  be  de- 
termined by  means  of  the  level  tester  shown  in  Fig.  19.  This 


^SS^S^^S^^^^^gg^^S:^ 

Fig.  19. 

consists  of  a  stout  cast  iron  frame  with  a  heavy  horizontal 
metal  bar  hinged  at  one  end  so  as  to  move  in  a  vertical  plane,  and 
supported  by  a  micrometer  screw  at  the  other  end.  By  one  turn 
of  the  screw  the  bar  may  be  tilted  through  an  angle  <£  whose  value 
in  radians  is  given  by 


(24) 

J-t 

where  p  is  the  pitch  of  the  screw  and  L  the  length  of  the  bar. 


32  PHYSICAL    MEASUREMENTS 

The  level  is  placed  upon  the  bar  and  the  readings  on  the  mi- 
crometer and  on  the  ends  of  the  bubble  recorded.  The  readings 
on  the  east  and  west  ends  of  the  bubble  in  the  direct  and  reversed 
positions  may  be  designated  by  e,  w,  e'  and  wf  respectively.  The 
screw  is  turned  until  the  bubble  moves  over  three  or  more  divi- 
sions and  the  readings  repeated.  Proceed  in  this  way  by  successive 
steps  of  from  three  to  five  divisions  throughout  the  length  of  the 
graduation.  Reverse  the  level  and  repeat  the  readings  for  the 
other  half  of  the  tube.  From  the  data  thus  obtained  compute  the 
value  of  in,  the  distance  the  bar  has  been  raised  to  move  the 
bubble  over  one  scale  division.  From  the  known  values  of  L  and 
m  we  may  at  once  compute  8,  the  angular  value  of  one  scale  divi- 
sion of  the  level,  from  the  relation 

•--f  M 

where  8  is  expressed  in  radians  per  division.     Reduce  this  value 
to  seconds  per  division. 

If  there  be  n  scale  divisions  graduated  on  the  tube,  then  n  8  is 
the  angle  subtended  by  this  part  of  the  tube  at  the  center  of  the 
circle  of  radius  R,  of  which  it  forms  a  short  arc.  If  the  length 
of  the  scale  be  /,  then  from  the  definition  of  an  angle  in  circular 
measure  we  have 

»»  =    -L.  (26} 

I  LI 

*  =  ^  =  7T^  (27) 

FORM  OP  RECORD. 

Exercise  p.  The  Level  Tester.  To  determine  the  value  of  a 
scale  diiision,  and  the  radius  of  curvature  of  a  spirit  level. 

Date 

L= /= 

Level    direct  Level   reversed 

Bubble  Screw  Bubble  Screw 


e 


w 


e 


w' 


Mean  value  of  m  = 


FUNDAMENTAL   MEASUREMENTS  33 

33.  Small  Angles  by  the  Filar  Micrometer.  Small  angles 
may  be  measured  with  great  accuracy  by  means  of  a  telescope 
of  large  focal  length  in  which  the  ordinary  eye-piece  has  been 
replaced  by  a  filar  micrometer. 

For  laboratory  experiments  the  telescope  should  have  a  focal 
length  of  at  least  50  centimeters  and  should  also  have  a  slow 
motion  in  azimuth.  A  large  reading  telescope  will  do  very  well. 
The  angle  to  be  measured  may  be  that  subtended  by  two  divisions 
of  a  brilliantly  lighted  scale  or  better  by  two  small  pin  holes  in 
a  piece  of  tin  foil  placed  before  a  small  lamp.  After  the  tele- 
scope has  been  sharply  focused  the  micrometer  is  revolved  in  the 
tube  until  the  transverse  hair  connects  the  two  points  in  ques- 
tion, in  order  that  the  distance  measured  may  be  the  actual 
distance  between  them.  By  the  slow  motion  screw  of  the 
telescope  the  fixed  hair  is  brought  upon  the  image  of  one  of  the 
points.  The  micrometer  head  is  then  turned  till  the  movable 
hair  coincides  with  the  fixed  hair  and  the  reading  noted.  The 
movable  hair  is  then  brought  upon  the  other  image  and  the  read- 
ing upon  the  head  again  recorded,  care  having  been  taken  to 
avoid  lost  motion  of  the  screw  in  making  the  settings  in  each  case. 

The  difference  between  the  two  readings  gives  the  angle  in 
terms  of  the  divisions  of  the  micrometer  head  and  the  focal 
length  of  the  telescope  objective.  Let  m  be  the  measured  dis- 
tance between  the  .images  and  F  the  focal  length  of  the  objective, 
both  expressed  in  millimeters  ;  then  <£,  the  angle  sought,  is  ex- 
pressed in  radians  by 

0=^.  (28) 

If  r  is  the  least  reading  of  the  micrometer,  then  <$>'  ,  the  least 
angle  that  can  be  measured  is 


(29) 


from  which  it  is  clear  that  the  delicacy  of  measurement  increases 
as  the  focal  length  of  the  objective  increases. 


34  PHYSICAL  MEASUREMENTS 

34.  Exercise  10.  Angular  Measurement  by  the  Filar  Mi- 
crometer. Take  a  piece  of  stiff,  dark  blue  card  board,  35  cms 
square,  and  cut  from  its  center  a  hole  5  cms  in  diameter.  Over 
this  paste  a  piece  of  rather  heavy  tin  foil  in  which  have  been 
pierced  two  small  holes  not  more  than  5  mms  apart.  Place  the 
card  board  in  front  of  a  fishtail  gas  burner  at  a  distance  L,  of  at 
least  20  meters  from  the  telescope.  The  bright  yellow  images  are 
seen  sharply  defined  upon  the  deep  blue  field.  Repeat  the  meas- 
urements at  least  five  times,  setting  the  fixed  hair  first  upon  one 
image  and  then  upon  the  other.  Return  the  final  value  of  the 
angle  in  seconds.  Measure  L  and  compare  the  measured  value 
with  the  computed  value  of  the  angle  subtended  by  the  holes  at 
the  center  of  the  objective.  Compute  also  the  least  angle  meas- 
urable with  the  apparatus. 


FORM  OF  RECORD 

Exercise  10.     The  Filar  Micrometer.     To  measure  the  angle 
subtended  by  two  points  distant meters  from  the  telescope. 


Date, 


1st    Image 


Readings 


2nd  Image 


Difference 


M  Mean  value  of  m 


$  0 

Observed  Computed        I      Difference 


35.  The  Sextant.  The  sextant  is  an  instrument  for  meas- 
uring the  angle  subtended  by  two  distant  points  or  the  angular 
elevation  of  a  point  above  the  horizon.  It  consists  of  a  light 
frame  (Fig.  20),  carrying  a  telescope  BF,  a  graduated  arc  AC, 
to  the  plane  of  which  the  axis  of  the  telescope  is  parallel,  and  two 


FUNDAMENTAL    MEASUREMENTS 


35 


20- 


mirrors  /  and  H,  whose  planes  are  normal  to  the  plane  of  the  arc. 

The  mirror  /,  or  index  glass,  is  attached  to  the  index  arm  IB, 

which  rotates  about  /  as  a 

center  and  carries  a  vernier 

along  the  arc  AC.   The  hor- 

izon glass  //,  is  fixed  in  po- 

sition and  is  silvered  on  its 

lower  half  and  transparent 

on  its  upper  half,  so  that  on 

looking    through     the    tele- 

scope,     objects      are      seen 

through   the   upper    half   in 

the   direction   HP,   while   in 

the  lower  half  are  seen  by 

reflection  objects  in  the  di- 

rection of  HP'. 

Let  it  be  required  to  meas- 
ure  the  angle  PHP'  subtended  at  H  by  the  points  P  and  P'.  The 
telescope  is  pointed  toward  P  and  sharply  focused.  If  the 
planes  of  the  two  mirrors  are  parallel  the  two  lines  of  sight  are 
parallel,  and  the  two  images  of  the  object  at  P  are  coincident. 
If  now  the  instrument  be  rotated  about  the  line  HP  as  an  axis 
until  the  plane  of  the  graduated  arc  passes  through  P'  ,  and  the 
index  arm  be  then  moved  so  as  to  bring  the  image  of'P'  also 
into  the  field  of  the  telescope,  the  two  images  may  be  made  to 
coincide.  In  this  case  the  light  from  P'  has  been  reflected  from 
the  two  mirrors  7  and  H,  and  consequently  it  has  been  deviated 
through  an  angle  equal  to  twice  the  angle  between  the  mirrors.1 
The  half  degree  marks  on  the  arc  are  numbered  as  whole  degrees, 
and  hence  the  reading  of  the  index  gives  directly  the  angle  PHPf. 

Figure  21  shows  the  usual  form  of  the  instrument,  in  which 
the  various  parts  will  be  recognized  -without  difficulty.  The  ver- 
nier arm  is  furnished  with  clamp  and  slow  motion  screw  for 
making  the  final  settings.  It  is  important  for  accurate  work  that 
the  light  from  the  two  objects  should  be  as  nearly  as  possible  of 
the  same  intensity.  To  this  end  shades  of  colored  glass  are 

1  College  Physics,  Article  442. 


36  PHYSICAL    MEASUREMENTS 

attached  to  the  instrument  which  may  be  rotated  in  front  of  either 
mirror  at  will. 

Adjustments  of  the  Sextant.  In  order  that  its  readings  may 
be  reliable  the  sextant  must  be  accurately  adjusted  and  carefully 
handled.  The  instrument  should  fulfill  the  following  conditions: 

(a)  The  plane  of  the  index  glass  should  be  normal  to  the 
plane  of  the  graduated  arc.  This  adjustment  is  secured  by  plac- 
ing the  eye  near  the  index  glass  and  adjusting  the  glass  until  the 
graduated  arc  and  its  reflected  image  appear  to  be  in  the  same 
plane. 

(&)  The  horizon  glass  should  also  stand  normal  to  the  plane 
of  the  graduated  arc.  This  adjustment  is  attained  when  the  two 


Fig.  21. 

images  of  a  distant  object  can  be  made  to  coincide  exactly. 

(c)  The  axis  of  the  telescope  should  be  parallel  to  the  plane 
of  the  graduated  arc.  This  adjustment  is  correct  when  the  images 
of  two  stars,  about  120°  apart  may  be  made  to  coincide  exactly 
on  either  of  two  cross  hairs,  equidistant  from  the  field  of  view. 

The  foregoing  adjustments,  when  once  carefully  made  should 
not  require  further  attention.  The  instrument  will  be  adjusted 
by  the  instructor  in  charge  and  beginners  are  cautioned  against 
attempting  such  adjustments  for  themselves. 

(d)  The  index  arm  should  read  zero  when  the  two  mirrors 
are  parallel.    This  adjustment  is  rarely  perfect,  and  when  made 


FUNDAMENTAL    MEASUREMENTS  37 

is  subject  to  frequent  changes.  It  is  therefore  better  to  correct 
for  the  error,  thus  introduced,  and  allow  for  it  in  all  readings. 
It  is  also  necessary  to  determine  this  correction  every  time  the 
sextant  is  used  as  it  varies  from  day  to  day. 

(c)  Index  correction.  The  correction  for  the  index  error  is 
determined  by  focusing  the  telescope  upon  some  distant  point  and 
bringing  the  two  images  into  exact  coincidence.  In  this  position 
the  mirrors  are  parallel  and  the  vernier  should  read  zero.  If  it 
does  not,  suppose  the  reading  to  be  A  ;  then  for  an  angle  which 
gives  the  scale  reading  B,  the  true  value  is  B — A.  In  some  cases 
the  value  of  A  is  negative,  or  the  arm  must  be  moved  back  be- 
yond the  zero  in  order  to  bring  the  images  into  coincidence.  For 
this  reason  the  graduation  is  continued  slightly  beyond  the  zero, 
as  shown  in  Fig.  21. 

36.  Exercise  n.     Measurement  of  Angles  by  the  Sextant. 

(a)  Measure  the  angle  subtended  by  two  distant  points. 

(b)  Measure  the  elevation  of  a  point  using  an  artificial  hori- 
zon. 

FORM   OF  RECORD 

Exercise  n.     To  measure  the  angle  betiveen  two  points  by  the 
sextant. 


Index  reading 


Mean 
Index    error. 


Angle 


Date. 


True  angle 


MASS 

37.  The  Balance.  Strictly  speaking  all  determinations  of 
mass  are  indirect.  The  balance  is  an  instrument  for  the  com- 
parison of  masses.  In  its  simplest  form  it  consists  of  a  light 
beam  turning  readily  about  its  middle  point  and  carrying  at  its 
ends  two  scale-pans  of  equal  weight.  When  disturbed  the  system 
oscillates  about  its  position  of  equilibrium  to  which  it  finally  re- 
turns. When  masses  are  placed  in  the  pans,  it  is  evident  that  the 
original  position  of  equilibrium  will  be  resumed,  only  when  the 
moments  of  the  forces  due  to  the  action  of  gravity  upon  these 


PHYSICAL    MEASUREMENTS 


masses  are  equal.  If  now,  we 
assume  the  arms  of  the  bal- 
ance to  be  equal,  we  may  set 
the  masses  equal  to  each  other 
when  their  moments  have  been 
shown  to  be  equal ;  i.  c.,  when 
the  balance  resumes  its  origi- 
nal position  of  equilibrium. 

In  balances  of  precision 
(Fig.  22),  the  beam  and  scale 
pans  are  hung  from  accurately 
ground  knife-edges  resting  up- 
on agate  plates.  When  not  in 
use,  the  knife-edges  are  re- 


Fig.  22. 


lieved  from  the  weight  of  the  pans  and  beam  by  means  of  an 
arresting  device.  This  must  always  be  used  when  weights  are 
to  be  changed,  or  articles  to  be  weighed  are  to  be  placed  upon 
the  scale  pans  or  removed  from  them. 

The  oscillations  of  the  balance  are  observed  by  means  of  a  long 
slender  pointer  moving  in  front  of  a  graduated  scale.  Care 
should  be  taken  to  raise  the  system  from  the  knife-edges,  only 
when  the  pointer  is  at  the  middle  of  the  scale.  A  system  of 
light  levers  is  usually  placed  under  the  pans,  to  maintain  the  bal- 
ance in  equilibrium  for  small  differences  of  weight  in  the  pans, 
and  to  prevent  undue  movements  of  the  beam  during  rough 
weighing.  The  beam  is  generally  divided  from  the  middle  out- 
ward, into  n  equal  divisions,  the  last  one  coinciding  with  the  knife- 
edge  over  the  pan.  By  sliding  upon  this  beam  a  small  wire  rider 
weighing  n  milligrams,  weighings  may  be  made  directly  to  mil- 
ligrams. For  subdivision  of  the  milligram  the  method  of  oscil- 
lations is  used. 

38.  Determination  of  the  Resting  Point.  Owing  to  the  loss 
of  time  incident  upon  waiting  for  the  balance  to  come  to  rest, 
it  is  usual  to  determine  the  final  position  of  the  pointer  from  a 
series  of  its  successive  turning  points.  Since  the  vibrations  of  the 
system  are  more  or  less  damped,  it  is  necessary  to  take  the  initial 
and  final  readings  of  the  pointer  on  the  same  side.  If  we  call 


FUNDAMENTAL   MEASUREMENTS  39 

the  central  division  of  the  scale  zero,  and  readings  to  the  right 
and  left  respectively  plus  and  minus,  then  the  resting  point  is 
found  by  averaging  the  means  found  for  each  side  separately? 
For  example,  if  the  readings  are 

Left  Right 

+  104 

—  9.2 

+  10.0 

-8.9 

+   9-7 


Mean  —  9.05  Mean  +10.03 

Resting  point  =   -  IO-°3     ,_  _j_  Q  4g 

This  means  that  the  pointer  would  finally  come  to  rest  at  a 
point  about  0.5  of  a  division  to  the  right  of  the  zero.  The  resting 
point  should  be  determined  both  before  and  after  making  a  weigh- 
ing, and  should  remain  constant  if  the  balance  be  properly  ad- 
justed. 

In  some  balances  the  scale  divisions  are  numbered  continuously 
from  left  to  right.  In  the  use  of  such  instruments  the  readings 
are  taken  directly  and  the  positive  and  negative  signs  are  avoided, 
a  method  which  seems  more  convenient  than  the  former.  If  the 
scale  divisions  are  not  numbered,  call  the  middle  mark  ten. 

39.  Sensibility  of  the  Balance.     If  the  rider  be  now  placed 
upon  the  first  division  of  the  beam  and  the  resting  point  deter- 
mined as  before,  then  the  difference  between  these  two  resting 
points  is  the  number  of  scale  divisions  through  which  the  beam 
has  turned  for  a  difference  in  weight  of  one  milligram.     This  is 
by  definition  the  sensibility  of  the  balance,  and  is  usually  termed 
the  sensibilty  for  zero  load  in  the  pans.     Since  the  sensibility 
varies  with  the  load,  it  is  always  necessary  to  determine  it  for  the 
specific  load  upon  the  pans. 

40.  Exercise  12.     To  Make  a  Single  Weighing.     A  single 
weighing  will  not  afford  an  accurate  determination  of  mass,  since 
the  equality  of  the  lengths  of  the  arms  is  tacitly  assumed,  and  this 
method,  though  commonly  used,  should  not  be  employed  in  meas- 
urements of  precision.     Moreover  it  does  not  require  a  smaller 
number  of  observations  than  the  much  more  precise  method  of 
double  weighing  described  below.    However,  where  only  relative 


.40  PHYSICAL    MEASUREMENTS 

weights  are  required,  as  in  determinations  of  specific  gravity,  or 
in  chemical  analysis,  single  weighing  is  permissible,  and  is  there- 
fore described  here. 

First  determine  the  resting  point  for  zero  load.  Next  place 
the  object  to  be  weighed  upon  the  left  hand  scale  pan  and  an  esti- 
mated equivalent  weight  upon  the  right  hand  pan.  Release  the 
arrest  very  slightly  and  note  the  indication  of  the  pointer.  If  the 
weight  be  too  small  it  should  be  sufficiently  increased  to  turn  the 
pointer  to  the  opposite  side  on  the  next  trial.  In  this  way  the 
true  weight  may  be  rapidly  approximated  to  the  nearest  gram, 
then  to  the  nearest  centigram.  After  this  the  balance  case  should 
be  closed,  the  rider  applied  and  the -pan  arrests  turned  down. 
Having  found  the  weight  to  the  nearest  milligram,  the  balance  is 
set  swinging  and  the  resting  point  is  determined.  The  rider  is 
then  shifted  one  whole  division  on  the  beam,  so  as  to  bring  the 
resting  point  on  the  other  side  of  the  zero  position,  and  the  rest- 
ing point  again  determined.  If  now  we  call  the  three  resting 
points  p0f  p,  and  P,  where  P  corresponds  to  the  weight  greater 
than  the  true  weight,  then  d  W,  the  fraction  of  a  milligram  to  be 
.added  to  smaller  weight  W ,  is 

(30) 

where  s  is  the  sensibility  for  the  given  load.  The  true  weight 
is  therefore 


Thus  suppose 

/»0  =  +  0-49 
p  =  +  0.77 
P  —  —  0.15 
then 

/>  —  />o      0.28 

J=P=W    =°-3  milligram. 

Care  should  be  taken  to  avoid  error  in  counting  up  the  weights 
upon  the  scale  pan.  An  excellent  method  is  to  write  down  the 
weights  from  the  vacant  spaces  in  the  box  and  then  check  each 
•weight  as  it  is  returned  to  the  box. 


FUNDAMENTAL   MEASUREMENTS  4! 


FORM    OP   RECORD. 

Exercise  12.    To  make  a  single  weighing. 

Balance  No Date. 

Object  to  be  weighed 

Approximate    weight    


P  —  Po= ;  S  =  P  —    = 


Weight  of —  W-\- 


41.  Reduction  to  Weight  in  Vacuum.     The  weight  of  a. 
body  in  vacuum  is1 

W  =  w(i+~—±')  (32) 

where  w  is  the  observed  weight  of  the  body,  d  its  density,  and 
a  and  D  the  densities  of  the  air  and  of  the  weights  respectively.  In 
weighing  a  quantity  of  water  with  brass  weights,  this  correction 
amounts  to  about  1.06  milligram,  for  every  gram.  It  is  there- 
fore imperative,  in  the  case  of  calibration  of  glassware  by  means 
of  water  or  in  similar  problems  to  make  the  reduction  to  vacuum. 

42.  Exercise  13.    Double  Weighing.2     In  order  to  eliminate 
any  inequality  between  the  two  arms  of  the  balance  the  object 
may  be  weighed  first  in"  the  left  hand  pan  and  then  in  the  right. 
The  true  weight,  WQ,  is  given  by  the  equation 

Wo  =  V  Wi  Wz, 
or,  since  for  every  good  balance  W \  and  W '2  are  very  nearly  equal, 

W.=  y2(Wl+W^.  (33) 

In  practice  the  operations  may  be  carried  out  as  follows :  Let 
'P0,  p,  and  p'  be  the  resting  points  for  zero  load,  load  on  the  left, 
and  load  on  the  right  respectively;  and  let  W  be  the  approxi- 
mate weight,  i.  e.,  the  weight  sufficient  to  balance  the  load  to 
within  one  milligram;  then  since 


1  College  Physics,  Article  76. 

2  For  detailed  treatment  of  the  balance  and  its  use,  see  Stewart  and 
Gee,  Practical  Physics,  Vol.  I,  pp.  63-94. 


42  PHYSICAL    MEASUREMENTS 

and   W,  =  W- 


(34). 

From  this  it  is  seen  that  in  double  weighing  the  determination 
of  p0  is  not  required.  The  sensibility  may  be  determined  with 
the  load  in  either  pan. 

It  will  be  instructive  to  determine  IV±  and  W2  separately  by 
single  weighing,  since  the  square  root  of  their  ratio  will  give  us 
the  ratio  of  the  lengths  of  the  arms  of  the  balance.  It  should  be 
noted  that  the  method  of  double  weighing,  as  described,  requires 
no  larger  number  of  observations  than  the  single  weighing  and 
is  far  more  accurate. 

FORM   OP   RECORD. 

Exercise  ij.  To  weigh  a  piece  of  brass,  by  the  method  of 
double  weighing. 

Balance  No  ...........  Date  .......... 

Object  to  be  weighed  .............. 

Approximate  weight,  W  = 

Load  left  Load  right  i  mg.  added 


p= 

—  p'= 


log  0.5  = 
log  (/»  —  /»')  = 
W  cologj  — 

log  A  W  — 

TIME. 

43.  Period  of  Vibration.  In  the  case  of  a  system  vibrating 
freely  about  its  position  of  equilibrium,  the  time  elapsing  between 
two  successive  passages  of  the  system  through  the  same  point  in 
its  path  in  the  same  direction,  is  defined  as  its  period  of  vibration. 
Most  measurements  of  time  in  the  physical  laboratory  consist  in 
determining  the  period  of  vibration  of  some  system,  as  a  pendu- 
lum, magnetic  needle,  galvanometer  needle,  etc.  The  problem 


FUNDAMENTAL    MEASUREMENTS  43 

presents  itself  most  frequently  as  the  determination  of  the  period 
of  a  pendulum,  and  although  what  follows  is  applied  directly  to 
this  end,  the  method  is  equally  applicable  to  the  case  of  any  freely 
vibrating  system. 

44.  Method  of  Coincidences.  The  process  usually  adopted 
is  a  modification  of  Borda's  method  of  coincidences.  The  essence 
of  the  method  consists  in  determining  the  time- necessary  for  the 
pendulum  to  complete  some  large  number  of  vibrations ;  from 
this,  if  the  number  of  vibrations  be  known,  the  period  of  a  single 
vibration  is  at  once  deduced.  Perhaps  the  simplest  way  is  to  note 
the  time  occupied  in  counting  100  or  1000  vibrations.  This  meth- 
od however,  is  both  tedious  and  inaccurate,  since  owing  to  its 
monotony  the  observer  is  liable  to  make  an  error  of  one  or  even 
of  ten  vibrations  in  counting  a  large  number.  To  this  source  of 
error  is  added  the  uncertainty  of  beginning  and  ending  the  count 
exactly  upon  the  second. 

In  order  to  avoid  the  first  source  of  error  the  pendulum  is  made 
to  keep  count  of  its  own  vibrations  when  compared  with  a 
clock  beating  seconds.  .  The  second  error  is  minimized  by  attach- 
ing a  pointer  to  the  pendulum  and  observing  its  passage  over  a 
scale,  or  in  work  of  greater  accuracy,  by  viewing  the  pendulum 
through  a  telescope  and  noting  the  time  of  its  passage  over  the 
cross-wires  in  the  focal  plane  of  the  eye-piece.  It  is  best  to  ob- 
serve this  passage  when  the  pendulum  is  in  the  middle  of  its 
swing,  and  moving  with  its  maximum  velocity.  The  clock  is 
connected  electrically  with  a  sounder  and  the  beats  are  thus  made 
audible  throughout  the  room. 

The  pendulum  to  be  timed  is  set  swinging  through  a  small  arc 
not  to  exceed  3°.  The  observer  at  the  telescope  notes  the  transits 
of  the  pendulum  image  from  left  to  right  over  the  cross-wires 
and  awaits  the  coincidence  of  such  a  transit  with  the  click  of  the 
clock.  He  then  notes  carefully  the  number  of  seconds  elapsing 
before  the  next  passage  of  the  image  over  the  cross-wires  in  the 
same  direction  and  estimates,  as  well  as  possible,  the  fraction  of 
a  second  in  tenths,  thus  gaining  a  roughly  approximate  period. 
Suppose  the  period  is  found  to  be  somewhere  between  2.3  and  2.5 
seconds. 


44 


PHYSICAL,   MEASUREMENTS 


A  coincidence  is  again  observed  and  the  seconds  counted  con- 
tinuously until  a  number  of  fairly  good  coincidences  have  been 
observed.  From  this  observation  a  closer  approximation  to  the 
period  can  be  obtained.  Thus  if  the  period  is  near  2.5  seconds, 
good  coincidences  will  occur  in  5,  10,  and  15  seconds,  i.  c.,  after 
2,  4  and  6  complete  vibrations  of  the  pendulum.  If  on  the  other 
hand  the  period  be  nearly  2.3  seconds,  then  fairly  good  coinci- 
dences will  be  noted  at  7  and  16  seconds,  and  a  good  one  at  23 
seconds,  the  intervals  corresponding  to  3,  7,  and  10  vibrations. 
The  imperfect  coincidences  at  7  and  16  seconds  are  due  of  course 
to  the  fact  that  the  interval  of  7  seconds  is  o.i  second  greater  than 
the  time  needed  for  3  swings,  and  that  of  16  seconds  is  less  by  o.i 
second  than  that  needed  for  7  swings  of  the  pendulum.  In  this 
way  it  is  possible  to  determine  the  provisional  period  accurately 
to  tenths  of  a  second.  The  observer  may  distinguish  between  the 
grade  of  coincidences,  by  underscoring  in  his  record,  a  mod- 
erately good  coincidence  with  one  line  and  an  excellent  one  with 
two. 

Several  trials  should  be  made  and,  if  necessary,  50  or  even  75 
seconds  counted,  in  order  to  determine  this  period  with  accuracy. 
Suppose  it  has  been  found  to  be  2.3  seconds.  The  observer  again 
awaits  a  good  coincidence,  noting  the  seconds  by  calling  each  one 
up  to  and  including  the  second  of  coincidence,  zero.  He  then 
walks  to  the  clock,  counting  the  seconds  as  he  goes,  and  on  the 
tenth  second  reads  the  time,  noting  the  seconds  first,  then  the 
minutes  and  then  the  hour.  This  recorded  coincidence  is  ob- 
viously ten  seconds  later  than  the  observed  one,  but  by  counting 
ten  seconds  each  time  before  reading  the  clock,  the  interval  be- 
tween the  coincidences  is  preserved  and  no  error  is  introduced. 

A  second  coincidence  is  observed  as  soon  as  possible  and  re- 
corded in  the  same  way.  Now  the  difference  between  the  first, 
and  second  readings  of  the  clock  gives  the  number  of  seconds 
corresponding  to  some  integral  number  of  swings  of  the  pendu- 
lum, and  a  glance  at  the  approximate  period  is  usually  sufficient 
to  show  what  this  number  is.  In  case  of  doubt  divide  the  time  by 
the  provisional  period  and  take  the  nearest  integer  as  the  number 
of  vibrations.  (Why?)  The  number  of  seconds  divided  bv  the 


FUNDAMENTAL    MEASUREMENTS  45 

number  of  vibrations,  gives  now  the  period  to  a  closer  degree  of 
approximation  than  before.  A  third  coincidence  is  observed  and 
recorded  as  before.  The  interval  between  this  coincidence  and 
the  first  corresponds  to  a  still  larger  number  of  integral  swings 
of  the  pendulum.  This  larger  number  is  found  by  dividing  the 
seconds  by  the  period  last  deduced,  and  the  new  value  of  the 
period  is  computed  as  before.  Thus  by  successive  observations 
we  find  intervals  corresponding  to  a  larger  and  larger  number  of 
vibrations,  using  in  each  case  the  period  last  found. 

In  this  way  the  period  of  a  pendulum  may  be  readily  and  rap- 
idly determined  to  thousandths  of  a  second.  After  a  little  practice 
the  student  is  able  to  judge  a  coincidence  accurately  to  o.i  of  a 
second.  Thus  in  the  given  example  after  20  minutes  of  observa- 
tion the  pendulum  would  have  made  something  over  500  vibra- 
tions, and  the  time  needed  for  this  number  of  vibrations  would 
be  determined  to  ±  0.2  of  a  second,  or  the  period  would  be  accu- 
rate to  thousandths  of  a  second. 

Example.     The  following  example  will  make  the  method  and 

computation  clear : 

SIMPLE  PENDULUM. 

February   8,   1899. 
Length,  130  cms. 
Good  coincidences,^,  14,  16,  23. 

Approximate  period  2.3  seconds. 

Coincidences                         Interval                No.  of  Period 

3  h.  19  min.  52  sec.                     in  seconds          Vibrations  in  seconds 

20     15             23           10  2.3 

20  38  46  2O  2.30O 

21  26  94  41  2.293 
(21     50)           118           52           2.269  ??. 

22  30  158  69  2.2S98 

23  57  245  107  2.28o7 

24  29  277  121  2.289" 

30  19  627  274  2.288s 

30  51  659  288  2.2881 

33  22  810  354  2.2881 

It  is  to  be  observed  that  the  period  T  gradually  approaches  a 

limiting  value  which  becomes  constant  to  thousandths  of  a  second 
as  soon  as  the  number  of  observed  vibrations,  n,  reaches  a  definite 
value.  If  the  maximum  error,  e,  made  in  taking  any  coincidence 
be  ±  o.i  of  a  second,  then  the  maximum  error  possible  in  any 
number  of  observed  seconds  is  0.2  of  a  second.  Hence  to  have 
the  period  T  constant  to  thousandths  of  a  second,  we  must  make 


46  PHYSICAL    MEASUREMENTS 

o.2/n  less  than  o.ooi,  or  n  must  be  greater  than  200.  Obviously  n 
must  be  larger,  the  larger  e  becomes.  How  large  must  n  be  taken 
if  e  =  zt  0.2  seconds  ? 

In  case  any  observation  gives  a  result  sharply  at  variance 
with  the  others,  the  difficulty  lies  either  in  the  arithmetical  work, 
or  in  a  false  reading  of  the  clock.  The  latter  error  renders  the 
observation  useless;  it  should  be  bracketed  as  indicated  above, 
and  the  next  taken  with  greater  care.  The  advantage  of  the 
method  is  that  no  single  error  in  reading  the  clock  can  perma- 
nently vitiate  the  result. 

Instead  of  determining  the  number  of  vibrations  by  dividing 
the  seconds  by  the  last  value  of  T  deduced,  it  is  much  simpler  to 
use  the  preceding  intervals  and  vibrations  as  measures  of  the 
new.  Thus  the  second  interval  46,  is  manifestly  double  the- first; 
hence  the  number  of  vibrations  must  be  twice  as  many,  or  20.  In 
the  third,  the  interval,  94  seconds,  is  twice  the  second  +  2  sec- 
onds; the  excess,  2  seconds,  corresponds  to  an  additional  vibra- 
tion; hence  n  =  2  X  20  +  I  =41.  In  the  seventh  determination 
the  interval,  277  seconds,  may  be  evaluated  for  n  in  a  number  of 
ways ;  for  example,  from  the  third  we  may  have,  after  multiply- 
ing by  three : 

sees.  vibs. 
282  =  123 
277 


277  =  121 

or  from  the  third  and  fifth  thus : 

94=  4i 
158=  69 


252=  no 
277 

+25=+!  i 

277  =  121 


For  the  measurement  of  small  intervals  of  time  the  tuning  fork 
furnishes  an  accurate  and  convenient  method.  For  the  practical 
application  of  this  method  see  subsequent  articles. 


FUNDAMENTAL   MEASUREMENTS 

45.     Exercise  14.    Period  of  Torsional  Pendulum. 


47 


FORM    OF   RECORD. 

Exercise  14.  To  determine  the  period  of  a  torsional  pendulum, 
to  o.ooi  of  a  second. 

Record  as  indicated  above,  p.  45. 

46.  Exercise  15.  The  Barometer.  The  barometer,  (Fig. 
23),  consists  of  a  closed  tube  of  glass  of  uniform 
bore  about  80  cms  long,  filled  with  mercury  and  in- 
verted in  a  dish  containing  mercury.  The  free  sur- 
face of  the  mercury  in  the  vessel  is  in  communica- 
tion with  the  outer  air.  In  the  cistern  barometer, 
the  reservoir  (Fig.  24),  has  a  bottom  of  leather 
which  is  adjustable  by  means  of  a  thumb- 
screw. A  small  ivory  point  extending 
downward  from  the  upper  surface  of  the 
reservoir,  forms  the  zero-point  from 
which  the  measurements  indicated  on  the 
scale  are  reckoned.  Before  reading  the 
barometer  the  mercury  in  the  cistern  must 
be  so  adjusted  by  means  of  the  screw  that 
the  surface  just  touches  the  ivory  point. 
The  upper  part  of  the  tube  is  then  gently 
tapped  to  free  the  mercury  surface  from 
the  sides  of  the  tube,  and  the  vernier  ad- 
Fig.  24.  justed  by  means  of  the  thumbscrew  at  the 
side,  until,  on  looking  through  the  slit  in  the  barom- 
eter case,  the  upper  part  of  the  meniscus  is  seen  to 
be  just  tangent  to  the  line  joining  the  sharp  edges  at 
the  front  and  back  of  the  vernier. 

In  the  instrument  made  by  Haak,  of  Jena,  we  have  an  auto- 
matic adjustment  of  the  mercury  in  the  cistern.  The  zero-point 
is  the  tip  of  a  vertical  tube  connecting  with  a  lower,  auxiliary 
reservoir.  Air  is  forced  into  the  lower  reservoir  by  means  of  a 
bulb,  thus  driving  mercury  into  the  reservoir  proper  and  covering 
the  zero  point.  On  releasing  the  bulb  the  mercury  flows  out  until 
the  tip  of  the  zero  point  tube  is  again  exposed ;  the  barometer  is 


Fig.  23. 


48  PHYSICAL    MEASUREMENTS 

then  in  adjustment  for  reading.  The  scale  is  etched  directly  upon 
the  tube  of  the  barometer,  and  fractions  of  a  millimeter  may  be 
read  with  ease  by  means  of  the  adjustable  ring,  at  the  top,  which 
carries  a  fine  line  for  subdividing  the  millimeter  divisions. 

Readings  on  the  barometer  must  be  corrected  for  temperature 
effects,  and  are  reduced  to  o°C  by  the  use  of  the  following 
formula  :x 

H,  =  Ht  [i —(a -&)*],  (35) 

where  H0  is  the  barometric  height  at  o°C;  Ht  is  the  barometric 
height  at  t°C;  a  is  the  coefficient  of  cubical  expansion  for  mer- 
cury, (01  =  0.000181  per  degree),  b  is  the  coefficient  of  linear  ex- 
pansion for  the  material  of  the  scale ;  ( for  glass,  b  =  0.0000085 
per  degree,  for  brass,  b  —  0.000019  per  degree). 

FORM    OF   RECORD. 

Exercise  15.  Adjust  and  read  each  barometer  three  times;  cor- 
rect for  temperature  and  compare  readings. 


Barometer  No.  i  Barometer  No.  2       Date. 

Least  count  Least  count  

Readings  Readings 


Mean    Mean    

Reduced  to  o°  C Reduced  to  o°  C 

Express  the  barometric  pressure  in  dynes  per  square  centimeter. 


1  For   reducing  the   barometric   reading  to   standard   conditions,   viz. 
o°  C,  sea  level,  in  latitude  45°,  we  have  the  complete  formula 


where  g  is  the  acceleration  due  to  gravity  at  the  place  of  observation  and 
#45  980.63  cm  per  sec.  per  sec. 


CHAPTER  II. 
ELASTICITY. 

47.  Definitions.  Elasticity  is  that  property  of  matter  by  virtue 
of  which  a  body  resists  the  action  of  a  force  tending  to  change 
its  shape  or  bulk,  and  resumes  its  original  shape  or  bulk  after  the 
force  is  removed.     If  a  body  possess  elasticity  of  shape  it  is 
called  a  solid ;  if  it  possess  no  elasticity  of  shape  it  is  called  a  fluid. 

Any  change  either  of  size  or  shape,  produced  by  the  action  of 
a  force  upon  an  elastic  body,  is  called  a  strain,  and  is  measured 
by  the  relative  change  produced.  The  reaction  against  the  de- 
formation is  called  a  stress  and  results  in  the  appearance  of  a 
force  resisting  further  deformation.  After  equilibrium  ensues 
the  applied  and  the  resisting  forces  are  equal.  Hence  stress  may 
be  evaluated  in  terms  of  the  applied  force  per  unit  area  of  the 
cross-section  upon  which  the  force  is  exerted.  In  the  metric 
system  the  unit  of  stress  is  the  dyne  per  square  centimeter. 

Fluids  possess  perfect  elasticity  of  bulk,  i.  e.,  they  return 
exactly  to  their  former  bulk  on  removal  of  the  compressing  force. 
Solids  do  not  all  recover  their  initial  shape  with  equal  promptness. 
In  some  cases  the  return  is  much  retarded,  especially  after  repeat- 
ed or  long  continued  distortion.  This  retardation  is  commonly 
termed  elastic  after  effect,  and  is  quite  noticeable  in  metals.  For 
every  solid  there  is  a  limiting  distortion  beyond  which  the  body, 
when  freed  from  the  distorting  force,  no  longer  completely  re- 
gains its  former  shape.  In  engineering  practice  the  elastic  limit 
is  usually  measured  in  terms  of  the  stress  producing  this  limiting 
distortion. 

48.  Hooke's  Law.    When  an  elastic  body  is  distorted  within 
its  limit  of  elasticity,  the  opposing  force  called  out  by  the  distor- 
tion, tending  to  restore  the  body  to  its  original  condition,  is  pro- 
portional to  the  distortion.     This  is  known  as  Hooke's  Law, 


50  PHYSICAL    MEASUREMENTS 

and  as  originally  stated,  "ut  tensio  sic  vis,"  expresses  the  pro- 
portionality between  the  distortion  and  the  restoring  force.  The 
applications  of  this  law  are  very  numerous  including  every  form 
of  elastic  reaction  against  strains  produced  by  external  agencies. 

49.  Coefficients   of  Elasticity.      In   general   twenty-one   co- 
efficients would  be  needed  to  express  completely  the  elastic  nature 
of  any  solid.    If  however,  the  solid  be  isotropic,  these  twenty-one 
coefficients  reduce  to  two :  the  coefficient  of  volume  elasticity,  e, 
and    the    coefficient    of    rigidity,    n.     The    general    expression 
for  these  coefficients  is  the  quotient  arising  from  dividing  the 
stress  by  the  strain. 

50.  Coefficient  of  Volume  Elasticity.    In  the  case  of  the  co- 
efficient of  volume  elasticity  e,  we  have  the  stress  measured  by 
the  applied  pressure  p,  divided  by  the  compression  produced, 
where  compression  denotes  the  change  in  volume  v,  divided  by 
the  original  volume  V ',  or 


(37) 


In  the  case  of  a  gas,  the  volume  is  at  all  times  a  function  of  the 
pressure  to  which  it  is  subjected.  Hence  for  gases  it  should  be 
noted,  that  the  coefficient  of  elasticity  is  to  be  defined  in  terms  of 
the  change  in  pressure  and  the  corresponding  change  in  volume. 

Since  these  changes  are  conceived  as  being  very  small,  if  we 
assume  a  volume  of  gas  V  ,  to  be  subjected  to  a  change  in  pressure 
dp,  producing  a  corresponding  change  in  volume  dV,  then  for  a 
gas, 


dV 


It  should  be  observed  that  the  expression  for  the  strain,  dV  /V  , 
denotes  a  dilatation  if  positive  and  a  compression  if  negative. 


ELASTICITY  51 

The  coefficient  ef  however,  has  reference  simply  to  the  absolute 
value  of  the  ratio  Vdp/dV,  and  is  therefore  independent  of  the 
sign.    The  coefficient  of  elasticity  of  volume  is  the  only  one  pos 
sessed  by  fluids,  and  is  of  special  interest  in  all  cases  involving 
the  propagation  of  disturbances  through  fluid  media. 

51.  Young's  Modulus.    In  solids  in  the  form  of  wires  or  rods, 
subjected  to  longitudinal  forces  tending  to  produce  either  elonga- 
tions or  compressions,  we  are  interested  in  the  relative  elongation 
or  compression  /,  produced  in  length  L,  and  cross  section  a,  by  a 
force  of  P  dynes,  when  the  body  is  free  to  contract  or  expand 
laterally.     In  general,  longitudinal  expansion  is  accompanied  by 
lateral  contraction  and  longitudinal  compression  by  lateral  dis- 
tention.     The  measure  or  modulus  of  the  elastic  behavior  of  a 
solid  under  such  conditions  is  known  as  Young's  Modulus,  and 
may  be  defined  as  the  ratio  between  longitudinal  stress  F/af  and 
longitudinal  strain  l/L ;  that  is, 

u       FL 

M=~aT  (39) 

or  Young's  Modulus  is  numerically  equal  to  that  force  which, 
when  applied  to  a  wire  of  unit  cross  section,  would  be  suffi- 
cient to  stretch  it  to  double  its  length,  provided  of  course,  that 
the  elongation  remained  proportional  to  the  force  at  all  times. 

52.  Simple  Rigidity.    Besides  the  elasticity  of  volume,  solids 
have,  as  we  have  seen,  elasticity  of  shape  as  well.     If  a  solid  be 
so  distorted  that  its  shape  alone  is  changed,  it  is  said  to  have 
undergone  a  shear.  Thus  if  we  conceive  all  the  particles  in  one 
plane  in  a  body  to  be  fixed,  and  all  the  remaining  particles  to 
move  in  planes  parallel  to  this  plane,  and  by  amounts  propor- 
tional to  their  distances  from  this  plane,  such  a  distortion  consti- 
tutes a  shear.     The  stress  caused  by  such  a  shear  is  called  a 
shearing  stress,  and  the  coefficient  of  rigidity  or  the  simple 
rigidity  is  the  quotient  obtained  by  dividing  the  shearing  stress 
by  the  shearing  strain. 

In  order  to  learn  how  these  quantities  may  be  experimentally 


PHYSICAL    MEASUREMENTS 


determined,  let  us  consider  a  circular  cylinder  (Fig.  25),  of  radius 
r,  held  vertically  by  a  rigid  clamp  at  the  upper  end  and  subjected 
to  a  torsional  twist  at  the  lower  end.  The  effect  of  such  a  twist 
is  to  produce  a  shearing  strain  throughout  the  cylinder.  Imagine 
the  cylinder  to  be  made  up  of  a  large  number  of  tubes,  one  inside 
the  other,  and  cut  at  right  angles  to  the  axis  into  a  large  number 
of  circular  sections.  Each  circular  section  would  be  composed  of 

a  large  number  of  concentric  rings. 
Now  the  shearing  strain  increases 
regularly  from  above  downward  from 
section  to  section,  and  when  the  low- 
er end  of  the  cylinder  has  been 
twisted  through  an  angle  0,  the  dis- 
tortion or  shear  for  the  outer  ring 
of  the  lower  section  will  be  the  arc 
r0,  and  the  shearing  strain  is  meas- 
ured by  the  ratio  of  the  shear  to  the 
distance  of  the  sheared  surface  from 
the  fixed  end,  or  by  rO/L ;  or  in  gen- 
eral, the  shearing  strain  at  any  point 
in  a  cylinder  is  the  circular  displacement  at  that  point  divided  by 
the  distance  from  that  point  to  the  fixed  end  of  the  cylinder. 

Again  since  the  cylinder  under  stress  is  in  equilibrium,  the  mo- 
ment of  the  couple  producing  the  shear  must  be  balanced  by  the 
moment  of  the  couple  called  out  by  the  shear;  or  by  Hooke's 
law,  it  must  be  proportional  to  the  shear  itself,  hence  also  pro- 
portional to  rO/L.  Now  by  definition,  the  simple  rigidity  n,  is 
the  proportionality  factor  between  shearing  stress  and  shearing 
strain;  hence  we  have  the  shearing  stress  =  nrO/L. 

Let  us  now  consider  the  entire  lower  circular  section  of  the 
cylinder,  and  in  that  section,  a  ring  of  radius  x  and  of  width  dx. 
The  shear  will  be  xQ/L,  and  the  shearing  stress  will  be  nxB/L. 
The  area  of  the  elementary  ring  is  2  ?r  x  dx,  hence  the  force,  due 
to  the  shearing  stress  and  equal  to  stress  times  area,  is 

2  TT  n  &  x3  dx 

a  r  =   ; 


This  force  acting  with  a  lever  arm  of  x,  gives  an  elementary  mo- 


ELASTICITY  53 

ment  of  shearing-  force   for  the  elementary  ring  of  width  dx, 
equal  to 

2,irnQxzdx 


(40) 

For  the  entire  section,  the  moment  of  the  shearing  force  will  be 
the  sum  of  the  elementary  moments  for  all  elementary  rings, 
whose  radii  vary  from  zero,  at  the  center,  to  r  at  the  surface  of 
the  cylinder,  and  so  the  moment  of  the  torsional  couple  called 
out  by  the  shear,  is  found  by  integrating  the  expression  in  equa- 
tion (40),  or 


=  r 

Jo 


2  TT  n  9  r* 
4L 

or 

JT  =  ^IH  (42) 

whence 

iL^- 

"=   --  (43) 


It  must  be  observed  that  0  is  here  expressed  in  radians.  If  0 
is  measured  in  degrees,  what  correcting  factor  must  be  intro- 
duced ? 

In  case  of  equilibrium  ^~is  equal  to  the  moment  of  the  tor- 
sional couple  producing  an  angular  twist  6,  in  a  cylinder  of  radius 
r,  and  length  L.  By  making  the  angular  twist  equal  to  unity 
we  have  the  moment  per  unit  angle 


and  finally,  by  reducing  the  length  L  and  the  angular  twist  0,  each 
to  unity  we  have 


*=  —  •  (45) 


54  PHYSICAL    MEASUREMENTS 

The  quantity  t  is  called  the  modulus  of  torsion  ;  it  is  numer- 
ically equal  to  the  moment  of  the  couple  required  to  produce 
unit  angular  twist  in  a  wire  of  unit  length. 

Summary:  —  We  have  now  defined  and  derived  expressions  for 
the  following  quantities: 

Coefficient  of  volume  elasticity    e  =  V  -777  (38) 

Young's   modulus..  M  =  -  r  (39) 

a  I 

Torsional   moment  ............     J^~  —  -  (42) 


Simple    rigidity  ...............         n  =  (43) 


Torsional  moment  per  unit  twist      <-/  —  -  -  —  (44) 


Modulus  of  torsion t  =  (45) 

In  the  experiments  which  follow  several  of  the  above  quantities 
will  be  measured  in  one  or  more  different  ways. 

53.  Exercise  16.  To  Verify  Boyle's  Law.  The  apparatus 
consists  of  a  cylindrical  reservoir,  (Fig.  26),  formed  of  a  glass 
tube  some  25  cm  long  and  3  cm  in  diameter,  into  which  is 
sealed  a  uniform  tube  B,  some  30  cm  long  and  12  mm  in  diam- 
eter, closed  at  the  top  by  a  square-cut  plug  carefully  cemented  in, 
and  at  its  lower  end  extending  to  the  bottom  of  the  larger  tube. 
At  the  lower  end  of  the  reservoir  is  sealed  on  a  second  tube  A, 
5  mm  in  diameter  and  about  200  cm  long.  This  tube  is  bent 
back  upon  itself  about  10  cm  below  the  reservoir,  so  as  to  be 
vertical  and  parallel  to  the  tube  B.  It  is  terminated  at  its  upper 
end  by  a  small  thistle  bulb,  for  convenience  in  filling  the  reservoir 
with  mercury.  The  instrument  is  mounted  upon  a  suitable  sup- 
port carrying  a  scale  graduated  to  millimeters  at  the  side  of  the 
tube  A,  throughout  its  entire  length.  At  the  bottom  the  scale 


ELASTICITY 


55 


fin 


A-- 


stands  between  the  two  tubes  so  that  readings  upon  the  height  of 
the  mercury  in  the  tubes  A  and  B,  may  be  made  from  opposite 
edges  of  the  same  scale.  A  small  side  tube 
is  sealed  to  the  reservoir  near  the  top  by 
means  of  which  air  may  be  forced  into  the 
reservoir  from  a  small  force  pump.  Be- 
fore beginning  the  experiment  the  instru- 
ment is  so  adjusted  that  the  shorter  tube 
B,  is  about  half  filled  with  mercury  when 
the  air  in  the  reservoir  is  at  atmospheric 
pressure.  Air  is  next  driven  in  through 
the  side  at  C,  by  means  of  the  pump, 
until  the  mercury  almost  fills  the  long 
tube  A.  The  air  in  B  is  now  under  a 
pressure  measured  by  the  barometer  col- 
umn plus  the  difference  in  height  of  the 
mercury  in  the  tubes  A  and  B,  and  is  cor- 
respondingly compressed. 

Varying  pressures  and  corresponding 
volumes  are  successively  secured  by  al- 
lowing small  quantities  of  air  to  escape 
through  the  tube  C.  Readings  are  made 
upon  the  height  of  the  mercury  in  A  and 
B,  and  upon  the  lower  end  of  the  plug  in 
B.  Readings  should  be  continued  until 
the  mercury  in  the  tube  A  falls  to  the 
level  of  the  mercury  in  the  reservoir.  The 
air  must  be  allowed  to  come  to  the  tem- 
perature of  the  room  after  each  setting 
before  the  reading  is  taken.  (Why?) 
If  it  be  desired  to  take  readings  at  pres- 
sures below  that  of  the  atmosphere,  the  pump  is  reversed  and  the 
air  partially  exhausted. 

According  to  Boyle's  law,  the  product  of  the  pressure  and  the 
volume  of  gas  is  a  constant,  for  a  constant  temperature1,       or 

P.V  =  C.  (46) 


1  College  Physics,  Article  77. 


PHYSICAL,    MEASUREMENTS 


By  definition,  the  coefficient  of  volume  elasticity  for  a  gas  is 


dP 
dV 


Also  from  (46)  by  differentiation,  we  get 


or 


dP 

dV 


(47) 


(48) 


Thus  we  see  that  for  a  perfect  gas  undergoing  isothermal 
changes,  the  coefficient  of  volume  elasticity  e,  is  at  all  times  equal 
to  the  pressure  P.  For  the  purpose  of  our  experiment  let  a  and  b 
denote  the  observed  heights  of  the  mercury  in  the  tubes  A  and  B. 
Let  P0  represent  the  atmospheric  pressure  at  the  time  of  the  ex- 
periment, p  =  a  —  b,  the  applied 
pressure,  and  let  V  represent  a 
quantity  proportional  to  the  result- 
ing volume  of  the  air  enclosed  in 
the  tube  B.  Then  equation  (46) 
becomes 


80_ 


60. 


4Q. 


3SL  ^ 


20 


40 


GO/ 


f 


.    . 


08 


or 


=  C/V. 


(49) 


(50) 


If  now  we  plot  the  observed 
values  of  p  on  the  Y  axis  and  the 
reciprocals  of  the  corresponding 
volumes  on  the  X  axis,  (Fig.  27), 
our  curve  is  represented  by  the 
'equation 

y  =  Cx  —  P«,  (51) 


Fig.  27. 


the  equation  of  a  straight  line  cut- 
ting the  Y  axis  at  a  point  P0  below  the  origin.  This  point  may  be 
considered  the  true  origin,  measured  from  which  the  values  of  y 
denote  the  successive  values  of  ef  corresponding  to  the  related 
values  of  V.  The  tangent  of  the  angle  included  between  the  curve 
and  the  X  axis  is  equal  to  C.  If  all  quantities  be  expressed  in  the 


ELASTICITY 


57' 


proper  units,  then  C  becomes  the  gas  constant.    Under  what  con- 
ditions does  e  approach  zero? 


FORM    OF   RECORD. 


Exercise  16.     To  verify  Boyle's  law. 
Reading  on  plug Barometer Date. 


Tube  A 


Tube  B 


colog  V 


Plot  values  of  p  and  \/V. .  Determine  the  value  of  P0  from  the 
curve  and  compare  the  result  with  the  barometer  reading  at  the 
time  of  the  experiment. 

54.  Exercise  17.  The  Jolly  Balance.  In  the  Jolly  balance 
the  elastic  body  is  a  spiral  spring  mounted 
upon  a  suitable  support  and  carrying  at  its 
lower  end  a  pan  to  receive  the  substance 
under  experiment.  To  this  pan  is  attached 
a  second  pan  to  carry  the  substance  when 
immersed  in  the  water.  The  support  con- 
sists of  two  telescoping  nickel-plated  tubes 
mounted  upon  an  adjustable  tripod  base. 
The  inner  tube  to  which  the  spring  is  at- 
tached, is  actuated  by  means  of  a  rack  and 
pinion  A  (Fig.  28),  and  is  graduated  in. mil- 
limeters at  its  upper  end  through  about  50 
cms  of  its  length.  A  vernier  $,  on  the  upper 
end  of  the  outer  tube,reading  to  tenths  of 
a  millimeter  gives  the  elongation  of  the 
spring  under  any  load  when  the  indicator  B 
attached  to  the  lower  end  reads  zero.  This 
indicator  consists  of  a  small  rod  of  alumi- 
num furnished  with  two  short  cross  arms, 
one  at  either  end  of  a  short  vertical  glass 
tube  enclosing  the  rod  and  supported  by  an 
adjustable  clamp.  The  glass  tube  is 
whitened  at  the  back  and  at  its  middle  has 
on  the  inside  a  fine  black  line  passing  en- 
tirely round  it.  The  aluminum  rod  carries  p^  2S. 


58  PHYSICAL    MEASUREMENTS 

at  its  center  a  small  cylinder  bearing  three  equidistant  black  lines, 
the  middle  one  of  which  is  made  to  coincide  with  the  line  on  the 
glass  tube  when  the  spring  is  brought  to  the  zero  position.  An 
adjustable  support  carries  a  small  vessel  containing  distilled  water. 

After  the  support  is  once  adjusted  it  should  remain  in  the  same 
position  throughout  the  experiment,  since  in  this  way  the  pan 
will  always  be  immersed  to  the  same  depth  in  the  water. 

In  use  the  support  is  adjusted  and  the  reading  r0,  taken  with 
both  pans  empty.  The  substance  under  experiment  is  then  placed 
in  the  upper  pan,  the  support  adjusted  and  a  second  reading  rlf  is 
made.  The  substance  is  then  transferred  to  the  lower  pan,  the  ad- 
justment made  and  the  reading  r2,  taken.  Then  for  the  density 
of  the  substance  we  have 


where  d  is  the  density  of  the  water  used.1 

Derive  and  explain  this  formula. 

For  substances  lighter  than  water  a  small  piece  of  metal  heavy 
enough  to  sink  the  substance,  is  placed  in  the  lower  pan  and  kept 
there  during  all  three  readings.  The  density  is  then  computed  as 
above. 

For  liquids  the  lower  pan  is  removed  and  a  suitable  sinker, 
usually  of  glass  is  attached  to  the  hook  by  a  fine  platinum  wire. 
Readings  are  made  with  the  sinker  in  the  air,  r0,  with  the  sinker 
immersed  to  a  definite  depth  in  the  water  r±,  and  with  the  sinker 
immersed  to  the  same  depth  in  the  liquid,  r2.  Derive  and  explain 
the  formula  in  this  case. 

It  is  necessary  to  exercise  care  in  the  use  of  the  apparatus  in 
order  to  secure  trustworthy  results.  Air  bubbles  must  be  re- 
moved from  the  substance  and  from  the  lower  pan  before  readings 
are  taken.  Determine  the  density  of  three  solids :  brass,  zinc  and 
paraffin,  and  of  a  solution  of  copper  sulphate.  Compare  results 
with  those  obtained  in  preceding  exercises. 


1  College  Physics,  Article  69. 


ELASTICITY  59 


FORM    OF   RECORD. 

Exercise  ij.     Density  by  means  of  Jolly  balance. 

Density  of Date, 


........  ........  ........  Density  ...... 

Mean  ........  ........  ........ 

55.  Exercise  18.  Young's  Modulus  by  Stretching.  An  iron 
bracket  firmly  attached  to  the  wall  near  the  ceiling  supports  a  long 
brass  wire  which  carries  on  its  lower  end,  by  means  of  a  clamp,  a 
cage  for  the  reception  of  weights.  Near  the  lower  end  of  the 
wire  is  attached  a  needle  upon  the  point  of  which  is  focused  a 
micrometer  cathetometer  reading  to  o.ooi  mm.  In  order  to  elimi- 
nate the  yielding  of  the  supporting  bracket,  a  small  rod  hung  from 
it  is  loosely  attached  to  the  wire,  and  bears  on  its  lower  end  a 
small  metal  square  or  flag,  near  the  needle  point,  so  that  both  flag 
and  point  appear  in  the  field  at  the  same  time  and  readings  may 
be  made  upon  them  successively.  An  adjustable  table  is  placed 
under  the  cage  before  weights  are  added  and  then  gradually 
lowered,  to  avoid  subjecting  the  wire  to  sudden  jerks.  A  weight 
of  two  kilograms  is  left  permanently  upon  the  cage  in  order  to 
free  the  wire  from  kinks  and  to  insure  uniformity  of  stretching. 

Readings  are  taken  upon  the  flag  and  the  point,  F0,  PQ,  with 
only  the  zero  load,  two  kilograms,  on  the  cage,  the  mean  of  three 
readings  being  used  in  each  case. 

Let 


The  table  is  then  raised  beneath  the  cage  and  two  kilograms 
added.  The  table  is  then  lowered  and  readings  made  as  before. 

Fn  / 

i  —  •  JT  i  —  k. 

In  this  way  readings  are  successively  taken  for  weights  of  two, 
four  and  six  kilograms  in  addition  to  the  zero  load.  After  this  the 
weights  are  removed,  two  kilos  at  a  time,  readings  being  made  as 
before  until  the  zero  load  remains.  From  the  values  thus  found 


6o 


PHYSICAL    MEASUREMENTS 


the  stretch  for  two  kilograms  is  computed  for  each  reading. 
from  the  reading  with  6  kilograms  added,  we  have 


Thus 


The  mean  of  the  values  of  /  thus  obtained,  is  taken  as  the 
stretch  produced  by  two  kilos.  Having  determined  L  and  a,  we 
have 


FL       2000  .  980  • 

M  =  —  r~  =  —     -  -  — 


dynes  per  cm. 


Determine  the  effect  upon  the  final  result,  (a)  of  an  error  ot 
I  mm,  made  in  the  measurement  of  L  ;  (&)  of  an  error  of  o.ooi 
mm,  made  in  the  determination  of  each,  /  and  r. 

From  a  preliminary  determination  of  these  three  quantities, 
compute  the  greatest  error  permissible  in  each  one  of  them,  if  the 
result  is  to  be  accurate  to  within  one  per  cent. 


L 

r 

Weights 


FORM    OF   RECORD. 

18.     Youngs  modulus  by  stretching. 

Date. 


Flag 


Point 


Computation : 
log  2000  = 
log  980  = 
log  L  = 
CO  log  7rr'= 
co log  /  = 


log    M  = 


Mean. . 


56.  Exercise  19.  Verification  of  the  Laws  of  Bending. 
The  bending  of  a  beam  supporting  a  weight  is  a  function  of  the 
weight  w,  supported,  of  the  three  dimensions  of  the  beam  /,  b,  d, 
and  proportional  to  a  constant  C,  which  depends  upon  the  material 
of  the  beam  and  the  manner  in  which  it  is  supported.  It  is  pro- 
posed in  this  exercise  to  determine  experimentally  these  relations. 


ELASTICITY  6 1 

Let  us  assume  for  the  purposes  of  the  investigation,  the  general 
expression  for  the  bending  B, 

B  =Cwa  I1   dy  <f  '  (53) 

where  a,  ft,  y,  e,  are  constant  exponents  to  be  determined  by 
experiment.  Since  this  expression  is  perfectly  general  it  will  in- 
clude all  possible  bendings  of  the  bar,  obtained  by  varying  the 
weight,  length,  breadth,  or  depth  of  the  bar,  either  successively  or 
simultaneously.  In  order  to  keep  clearly  in  mind  the  relations 
under  investigation  it  is  best  to  vary  but  one  of  these  quantities  at 
a  time,  and  observe  the  bendings  produced  under  definite  con- 
ditions. Thus  with  a  bar  of  definite  length,  breadth  and  depth, 
vary  the  load  successively  and  observe  the  bendings  Blf  B2,  B3, 
B±,  B5,  produced  by  the  loads  zvlt  wz,  w3t  zv4  ws. 

Inserting  related  pairs  of  values  of  B  and  w,  as  B±  and  w^  B2 
and  zc'o  in  the  general  formula  we  have 


B2  =  Cwl'    dyd     ,  and  so  forth. 
Passing  to  logarithms  we  have 

log  5i  =  log  C  -f  a  log  zt/i  +  jS  log  /  -f-  7  log  b  +  e  log  d 
log  B2  =  log  C  +  a  log  ws  -|-  P  log  /  -f-  7  log  b  -f-  e  log  d 

log  Bi  —  log  B2  =  a  (log  wi  —  log  ze/2), 
or 

.     log  Bx- log  A 

n~    log™ -log**  (54) 

Where  a12  denotes  that  this  particular  value  of  a  is  derived 
from  the  related  values  of  B^,  «/±  andj52,  w2. 

Again  by  using  a  constant  difference  in  load,  w,  and  varying 
the  lengths  /,  we  obtain  a  series  of  bendings  B\,  B\,  B'z,  B\ 
B'r,,  corresponding  to  the  lengths  Ilt  12,  13,  I4f  /5.  Applying  the 


62 


PHYSICAL    MEASUREMENTS 


general  formula  to  the  values  of  the  bendings  B\  and  B'2,  for 
lengths  /j  and  12,  we  have,  in  the  same  way  as  before, 

log  B\  =  log  C  -f  a  log  w-{-p  log  /i  -f  7  log  b  -f  e  log  d 
log  5'2  =  log  C  +  a  l°g  w  +  jS  log  /2  +  7  log  b  -f  e  log  rf 


from -which 


log B\— - 
tog/a-log/ 


(55) 


Similarly  by  determining  the  bending  for  a  definite  weight  in 
bars  of  the  same  metal,  and  having  /  and  d  the  same,  but 
varying  the  breadth  b}  we  find  for  the  successive  breadths  b^  b2, 
&c.,  the  corresponding  bendings  B'\,  B"2,  &c.,  and  deduce  the 
relation 

'"-'^  (56) 


(57) 


?t  —  lOg  &2 

and  finally  by  varying  the  depth,  we  have 

log  J3ff/i  — logg^ 
12      log  di  —  log  d2 


The  apparatus   (Fig  29),  consists  of  a  graduated  scale  upon 
which  slide  two  knife-edge  supports  for  the  metal  bar,  one  of 

which  is  connected 
to  a  battery.  At 
the  middle  of  the 
graduated  scale  is 
mounted  a  microm- 
eter screw  whose 
point  rests  upon  the 
back  of  a  knife- 
edged  stirrup,  sit- 
ting  upon  and  at 
right  angles  to  the  bar  at  its  middle  point,  and  carrying  the  pan  in 
which  the  weights  are  placed.  The  micrometer  screw  is  connected 
to  the  other  pole  of  the  battery  and  the  circuit  is  completed  through 
the  bar  itself  and  the  stirrup.  A  telephone  receiver  placed  in  the 


-  29- 


ELASTICITY  63 

battery  circuit  gives  notice  of  contact  between  the  point  of  the 
screw  and  the  stirrup. 

In  use  the  supports  are  placed  at  equal  distances  on  each  side 
of  the  micrometer  screw,  and  the  bar,  with  the  stirrup  on  it,  is  so 
placed  upon  the  supports  as  to  leave  equal  lengths  projecting 
over  the  supports  at  each  end.  The  bar  is  brought  directly  under 
the  screw  point  and  the  stirrup  placed  accurately  at  right  angles 
to  the  bar  and  under  the  screw.  The  pan  containing  the  weights 
must  hang  clear  of  all  objects.  The  apparatus  having  been  per- 
fectly adjusted  the  length  of  the  bar  I,  between  the  supports,  is 
read  and  recorded.  The  micrometer  screw  is  then  slowly  turned 
down  until  the  telephone  gives  notice  of  contact  between  the  screw 
and  the  stirrup.  The  screw  is  then  carefully  turned  back  until  the 
snap  due  to  breaking  the  circuit  is  heard  in  the  telephone.  The 
zero  reading  is  then  taken,  using  the  mean  of  three  in  each  case. 

In  place  of  the  battery,  telephone  and  micrometer  screw,  we 
may  set  one  end  of  an  optical  lever  upon  the  stirrup,  the  two 
middle  legs  resting  upon  a  fixed  support  near  the  bar,  and  take 
readings  by  means  of  a  telescope  and  vertical  scale.  The  absolute 
value  of  the  readings  of  the  apparatus  may  be  determined  once 
for  all  by  depressing  the  middle  of  the  bar  by  means  of  a  microme- 
ter screw  and  noting  the  related  scale  readings. 

For  the  purposes  of  this  exercise  however,  all  that  is  necessary 
is  the  relative  depression  for  the  different  conditions,  and  these 
are  given  by  the  scale  readings.  This  method  has  the  combined 
advantages  of  greater  speed  and  greater  accuracy  in  making  the 
readings. 

For  this  experiment  four  bars  of  brass  of  rectangular  cross- 
section  and  about  70  cms.  in  length  are  employed.  One  of  the 
transverse  dimensions  is  the  same  for  all  bars  so  that  when  placed 
with  this  dimension  vertical,  we  have  a  series  of  bars  of  constant 
depth.  The  other  transverse  dimension,  for  the  four  bars,  differs 
in  each  case,  so  that  when  this  dimension  is  placed  vertical  we 
have  a  series  of  bars  of  constant  breadth  and  variable  depth. 

To  determine  the  bending  due  to  100  grams,  or  to  any  weight 
whatever,  it  is  better  to  take  the  zero  reading  with  an  initial 
weight  in  the  pan.  Then  to  find  the  bending  for  any  given  load, 


64  PHYSICAL    MEASUREMENTS 

as  100  grams,  the  bending  is  taken  for  the  initial  load  plus  the 
100  grams,  and  the  difference  between  this  reading  and  the  initial 
reading  gives  the  bending  for  100  grams.  A  new  initial  load  is 
taken  and  the  same  process  is  repeated,  thus  giving  a  second  value 
for  the  bending  for  100  grams ;  a  third  initial  load  is  chosen  and  a 
third  value  for  B  is  determined.  The  mean  of  the  three  values 
thus  found  is  the  bending  for  100  grams.  This  method  is  to  be 
pursued  in  rinding  the  bending  for  any  weight  whatever. 


FORM    OF   RECORD. 


Bending  for  100  grams  Date , 

Bar  No Length 

Load  Readings  Means  Difference  for  100  grams 


50  g. 


ISO  g. 


75  g- 


100  g. 
200  g. 


Mean  bending  =  B  for  100  g 


The  exercise  may  now  be  completed  under  the  following  heads : 
(ai)  Vary  w  and  determine  B  for  five  different  loads.  Use 
as  loads  the  weights  75,  100,  150,  200,  and  250  grams.  Use  a 
single  bar  about  60  cms  in  length.  Do  not  at  any  time  exceed  300 
grams.  Determine  Blt  B2,  B3,  B±,  B5,  for  w^,  w2,  «r3,  W4,  «/..  Ap- 
ply formula  (54),  combining  the  observations  two  by  two.  The 
mean  of  the  ten  values  of  a  obtained  in  this  way  is  taken  as  the 
value  of  a  to  be  substituted  in  the  general  formula.  In  case  any 
value  of  B  as  Bn,  has  been  wrongly  determined,  then  every  value 


ELASTICITY 


of  a  containing  the  subscript  n  will  differ  sharply  from  the  rest 
and  will  indicate  that  the  nth  observation  should  be  taken  with 
greater  care.  A  glance  at  the  tabulated  values  of  a  will  generally 
reveal  any  such  pronounced  error  of  observation. 


FORM    OF   RECORD. 

Exercise  ip.     Verification  of  the  laws  of  bending. 

First  part  as  given  on  preceding  page.  Date , 

Second  part  thus : 


B 

cms. 

log  B 

iff 

grams 

log  w 

Bi 

Wi 

B2 

w* 

and  so  on. 


log  Bi  —  log  £2  = 
log  Bi  —  log  B3  = 


log  wi  —  log  «k 

log  Wi  —  •  log  wz 


(b)  Vary  I  and  determine  B'  for  a  constant  load  w—  100 
grams.  \  For  lengths  30,  40,  45,  50,  55  cms,  determine  B\,  B'2, 
B'3f  B\,  B'5  for  the  constant  load  w  =  100  grams.     Apply  form- 
u^a  (55)>  combining  the  observations  as  in  (a).     Take  as  the  final 
value  for  ft  the  mean  of  the  ten  values  found  as  above.     Record 
as  in  (a),  substituting  /  for  w. 

(c)  Vary  b  and  determine  B",  for  a  constant  load  of  150 
grams.     Use  bars   i,  2,  3,  4,  with  constant  depth.     Measure 
bi,  b<2,  b3J  b±,  by  means  of  the  micrometer  gauge.    Determine  for 
bi,  b.2,  b3,  b±,  the  corresponding  values  B'\,  B"2,  B"z,  B'\.     Apply 
formula  (56),  computing  as  in  (a)  and  (b).     Record  as  before. 

(d)  Vary  d  and  determine  B  for  a  constant  load,  w=i$o 
grams.     Use  bars  with  constant  breadth.     For  depths  dl}  d2,  ds, 
d±,  determine  the  bending  £/",  BJ"  Bz"',  B^f.     Compute  from 
formula  (57).     Record  as  before. 

O)  Insert  final  values  of  a,  fi,  y,  e,  in  the  general  formula. 
Formulate  the  laws  of  bending  in  words. 

Curves:  The  observations  may  be  graphically  represented  by 
curves.  In  cases  of  direct  proportionality  between  the  bending 
and  the  quantity  which  was  varied  in  the  experiment,  plot  this 


66  PHYSICAL,    MEASUREMENTS 


quantity  as  abscissa  and  the  corresponding  bending  as  ordinate. 
The  curve  should  be  a  straight  line.  If  the  exponent  of  the  vari- 
able quantity  is  not  unity  a  straight  line  may  be  obtained  by 
plotting  the  logarithms.  Plot  the  curves  for  each  of  the  four 
series  of  observations. 

57.  Exercise  20.  Young's  Modulus  by  Flexure.  As  we 
have  learned  in  the  last  exercise,  the  bending  of  a  bar  of  length 
I,  breadth  b,  and  depth  d,  under  a  load  w,  is  expressed  by  the 
equation 

*-*££:  (58) 

It  has  already  been  observed  that  the  constant  C,  depends  upon 
the  mode  of  support  and  the  material  of  which  the  beam  is  com- 
posed. It  is  shown  in  the  theory  of  elasticity  that  when  the  beam 
is  supported  by  its  ends  the  value  of  the  constant  relating  to  its 
mode  of  support,  is  1/4,  and  when  supported  from  one  end  its 
value  is  4.  There  remains  therefore,  the  undetermined  part  of 
our  constant  C,  depending  upon  the  material  of  the  beam.  It  may 
be  shown  from  mathematical  analysis  that  this  part  of  the  con- 
stant is  \/M,  where  M  is  Young's  modulus  for  the  material  in 
question.  If  the  bar  be  supported  at  the  ends  as  usual,  then  we 
ma  write1 


B== 

or 


If  now  we  insert  in  this  formula  the  dimensions  of  bar  No.  i, 
we  shall  find  for  any  bending  B  and  the  related  load  w,  a  value  for 
M  very  nearly  that  obtained  in  Exercise  18  as  Young's  modulus 
for  the  material  of  the  bars. 


1  It  may  'be  shown  directly  that  the  expression  for  M  as  given  above, 
is  true  for  all  bars  supported  at  the  ends  and  loaded  at  the  middle.  Such 
proof  would,  however,  exceed  the  limits  set  for  this  text.  See  Stewart  and 
Gee,  Vol.  I,  pp.  162-195. 


ELASTICITY  67 

Moreover  it  appears  that  Young's  modulus  is  concerned  here  if 
we  consider  attentively  what  takes  place  in  the  bending  of  a  bar. 
We  see  that  upon  the  under  side  the  bar  is  stretched,  while  upon 
the  upper  side  a  compression  must  ensue.  Now  the  resistance  to 
this  stretching  stress  on  the  one  side  and  to  the  compressing  stress 
on  the  other,  form.  the  two  parts  of  a  couple  tending  to  right  the 
bar  under  its  load,  and  the  stress  divided  by  the  strain  gives 
again  Young's  modulus  for  the  material  in  question. 

FORM   OF  RECORD. 

Exercise  20.  .  .  To  determine  Young's  modulus  of  brass  and  steel 
by  the  method  of  flexure. 

I.  For  brass.  Date  ............ 

From  Exercise  19  (a) 

w  =  ......  log  w  =  ...... 

/  =  ......  log  P  =  ...... 

b  =  ......  colog  4  =  ...... 

d  =  ......  colog  b  =  ...... 

B  =  ......  colog  ds=  ...... 

colog  B  =  ...... 

M=  ......  - 

logM  =  ...... 

II.  For  steel. 

(a)  First  part  as  on  page  64. 

(b)  Computation  as  above. 

58.  Exercise  21.  Simple  Rigidity.  As  shown  in  equation 
(43)  the  expression  for  the  simple  rigidity  of  a  cylinder  under 
torsional  stress  is 


where  ^ls  the  moment  of  the  torsional  couple  needed  to  produce 
an  angular  twist  of  6  radians  in  a  circular  cylinder  of  length  L 
and  radius  r.  In  order  to  determine  n  we  must  measure  the  four 
quantities  ^\  B,  L,  and  r.  This  is  most  readily  done  by  means  of 
the  following  apparatus. 

A  metal  rod  (Fig.  30)  some  150  to  200  cm    in  length,  is  sup- 
ported horizontally  and  held  firmly  at  one  end  by  a  clamp.     The 


68 


PHYSICAL    MEASUREMENTS 


other  end  is  held  clamped  in  the  axis  of  a  light  aluminum  wheel, 
some  20  cm  in  diameter,  which  is  mounted  on  ball  bearings,  and 
graduated  on  its  face  through  180°.  The  wheel  is  furnished  with 
a  double  vernier  for  reading  its  position  at  any  time,  and  carries 
on  its  rim  a  flat  steel  tape  to  which  is  attached  the  holder  for  the 
weights  producing  the  torsion. 

The  manipulation  is  as  follows :  A  zero  load  of  some  50  grams 
is  first  placed  in  the  pan  and  the  zero  readings  taken.  A  load  of 
loo  grams  is  then  added  and  the  reading  again  taken.  The  differ- 


Fig.  30. 


ence  between  the  two  readings  is  the  angular  twist  for  100  grams. 
The  load  in  the  pan  is  again  increased  by  100  grams  and  the  read- 
ing made.  Half  the  difference  between  this  reading  and  the  first 
is  likewise  the  twist  for  100  grams.  The  load  in  the  pan  is  in- 
creased in  this  way  by  successive  steps  of  100  grams  until  500 
grams  have  been  added  and  then  diminished  by  similar  steps  until 
the  zero  load  remains.  The  twist  for  100  grams  is  computed  from 
each  reading  by  combining  it  with  the  zero  reading  as  shown 
above.  The  mean  of  the  values  thus  found  is  the  angular  twist  6, 
expressed  in  degrees,  for  a  load  of  100  grams.  The  length  of  the 
rod  L,  and  the  radius  of  the  wheel  /,  are  to  be  determined  by 
measurement.  Determine  the  radius  of  the  rod  from  five  measure- 
ments of  the  diameter  by  means  of  the  micrometer  gauge. 


ELASTICITY 


FORM    01?   RECORD. 


Exercise  21.    To  determine  the  simple  rigidity  of  two  metals. 


Date 

Computation  : 

Load 

Readings 

Diff.    for 

360  .  L  .  m  .  g  .  I 

CQ  erns 

100  gms. 

log  1,60  —  .  . 

i  TO    " 

r    

,  °  °  r        

log   L  —  .  . 

2^0    " 

M-  — 

100      log  tn  —  

-2CO        " 

a  — 

080      log   a    - 

Mean 

T= 

log    /    — 

0  — 

.  colog    0    —  

7'  — 

"olog  it*  —  

colog  r*    =  

log  n    — 

n   —  .  . 

59.  Exercise  22.  Simple  Rigidity  of  a  Brass  Wire  from 
Torsional  Vibrations.  We  have  learned  from  Hooke's  law  that, 
within  the  limits  of  elasticity,  the  restoring  force  called  out  by 
any  distortion  is  simply  proportional  to  that  distortion.  An  im- 
portant consequence  of  this  law  is,  that  if  a  heavy  body  be  sus- 
pended by  a  wire  and  the  wire  be  twisted  through  a  moderate 
angle  and  then  released,  the  restoring  force  is  continually  propor- 
tional to  the  distortion.  The  motion  induced  is  simple  harmonic 
and  consequently  the  vibrations  of  the  body  are  isochronous. 
Now  if  /  be  the  moment  of  inertia  of  the  body,  and-^the  moment 
of  the  torsional  couple  divided  by  the  angular  twist,  then  T,  the 
time  of  a  complete  vibration  of  the  system,  is  given  by  the  equation 


from  which 


(60) 


(61) 


We  have  also  seen   from    (44)    that  the  coefficient  of  simple 
rigidity  is  defined  by  the  equation 

TJ*  ft  Y 

c^r=:"HT 

whence,  equating  values  for  ^"ancl  solving  for  n,  we  have 

»=*#.  (fe> 

an  expression  involving  only  quantities  amenable  to  measurement. 


70  PHYSICAL,    MEASUREMENTS 

The  apparatus  consists  of  a  heavy  metal  cylinder  suspended  by 
a  brass  wire  and  furnished  with  a  light  pointer  moving  over  a 
horizontal  circular  scale.  By  means  of  an  adjustable  clamp  the 
length  of  the  torsional  pendulum  may  be  varied  within  wide 
limits.  The  times  of  vibration  7\,  T2,  Ts,  for  lengths  Llf  L2,  L3, 
are  determined  to  thousandths  of  a  second  by  the  method  given 
in  Article  44.  The  moment  of  inertia  I,  of  the  cylinder  is  com- 
puted from  its  mass  M,  and  radius  R.  The  radius  of  the  wire  is 
determined  by  means  of  the  micrometer  gauge.  The  insertion  of 
any  pair  of  related  values  of  T  and  L  in  formula  (62)  gives  a 
value  for  n.  Compute  the  three  values  and  return  the  mean  as 
the  value  found  for  n. 

FORM   OF  RECORD. 

Exercise  22.     Simple  rigidity  of  brass  wire  from  torsional  vibra- 
tions. 

Date  .................. 

First  part  as  given  under  Exercise  14  Computation  : 

Second  part  thus:  8*1  L 

— 


log 


M  — log  / 

R= log  L 

I  •=. colog  r* 

r= colog  T2 

log  n 

M 


CHAPTER  III. 

PENDULUM  EXPERIMENTS  AND  MOMENT  OF  INERTIA. 

60.  The  Simple  Pendulum.  For  practical  purposes  the 
pendulum  may  consist  of  a  lead  ball  about  3  cm  in  diameter  sus- 
pended by  a  fine  steel  wire  some  three  meters  in  length.  The  wire 
is  supported  from  the  upper  end  of  a  substantial  wooden  bar  two 
and  a  half  meters  in  length  about  10  cm  wide  and  5  cm  thick, 
fastened  firmly  to  the  wall  in  a  vertical  position.  The  bar  has 
let  into  its  front  face  a  brass  strip  ruled  to  millimeters  and  read- 
ing continuously  throughout  its  length  of  250  centimeters.  Over 
this  bar  slides  a  wooden  clamp  which  may  be  set  at  any  position, 
and  carries  on  its  front  a  short,  horizontal  brass  rod  10  cm  long 
and  i  cm  in  diameter,  furnished  at  its  outer  end  with  a  diametral 
slit  set  vertically,  and  adjustable  by  means  of  a  set  screw.  In 
this  way  the  fine  steel  wire  of  the  pendulum  may  be  clamped  in  the 
slit  of  the  rod  at  any  point  and  the  length  of  the  pendulum  varied 
between  50  and  300  centimeters. 

Such  a  pendulum  approximates  very  well  an  ideal  simple  pen- 
dulum. If  h  be  the  distance  from  the  lower  side  of  the  clamp  slit, 
to  the  center  of  the  ball  and  r  the  radius  of  the  ball  then  the  length 
I,  of  the  equivalent  ideal  simple  pendulum  is  given  by 


(63) 


Compute  the  error  produced  by  omitting  the  last  term  of  this 
formula  in  the  above  pendulum,  when  h  is  100  cm. 

61.     Exercise  23.     Law  of  the  Simple  Pendulum.     The  ob- 

ject of  this  exercise  is  to  investigate  the  relation  between  the  period 
of  a  simple  pendulum  and  its  length.  Since  the  most  casual  ob- 
servation shows  that  the  period  of  a  pendulum  is  some  function 


PHYSICAL    MEASUREMENTS 


of  its  length,  we  may  assume  as  a  general  expression  for  the  exist- 
ing relation 

(64) 


"where  C  and  m  are  constants  to  be  determined  by  experiment. 
Passing  to  logarithms  and  solving  for  m,  as  in  Exercise  19,  we  find 
for  in  the  value 


log  T3  —  log  Ti 
log  la  —  log/! 


(65) 


From  a  series  of  observations  on  five  pendulums  of  different 
lengths  we  get,  by  combining  as  in  Exercise  19,  ten  values  of  m. 
The  mean  value  of  m  so  determined,  when  inserted  in  the  equa- 
tions connecting  related  times  and  lengths,  gives  five  independent 
equations  for  C,  of  the  form 


log  C  =  log  T  —  m  log  /. 


(66) 


The  mean  value  of  C  thus  determined,  and  the  mean  value  of  m 
when  inserted  in  equation  (64),  give  the  relation  sought.  The 
•exercise  is  to  be  completed  as  follows  : 

(cc)  Determine  to  thousandths  of  a  second  the  period  of  vibra- 
tion of  a  pendulum  for  five  lengths  120,  140,  160,  180  and  200 
cms. 

(&)  Compute  from  these  observations  the  values  of  m  and  C 
as  described  above,  observing  the  arrangement  adopted  on  page 

(65). 

(c)     Plot  curve,  using  values  of  log  T  as  ordinates  and  those 
of  log  /  as  abscissae. 


FORM  OF  RECORD. 


Exercise  23.     To  determine  the 


(«)  Use  form  of  record  given  under  Art.  44. 
(&)         /  log  /  T  log  T  m 


of  the  simple  pendulum. 

Date 

C 


PENDULUM    EXPERIMENTS 


62.     Exercise  24.     Computation  of  g.     From  the  well  known, 
formula  for  the  time  of  vibration  of  a  simple  pendulum 


2\^ 


we  see  that 


or 


C  = 


2  7T 


47T 


(67) 


(68) 


(69) 


From  the  mean  value  of  C  found  in  Exercise  23,  compute  the 
value  of  g.  The  value  of  g  found  at  a  place,  H  meters  above  sea 
level,  may  be  reduced  to  sea  level  by  means  of  the  equation 


g0  —  g  -j-  0.0003086  H 


(70) 


Compare  this  reduced  value  g0  with  the  normal  value  y0  at  sea 
level  and  latitude  <j>,  given  by  Helmert's  formula 


70  =  978.03  (i  -j-  0.005302  sin  0  —  0.000007  sin2  <f>  ) 


FORM  OF  RECORD. 

Exercise  24.    To  compute  value  of  g. 


C  — 
log  C  = 


Date. 


log  4  = 

log  7T2  — 

colog  C'  = 


log  flr  — 


9  — 


63.     Exercise  25.     Moment  of 'Inertia  of  a  Connecting  Rod.. 

The  moment  of  inertia  of  a  body  about  its  center  of  gravity  may 
easily  be  determined,  provided  the  body  can  be  vibrated  as  a 
physical  pendulum  from  a  point  whose  distance  from  the  center 
of  gravity  is  accurately  known. 


74  PHYSICAL    MEASUREMENTS 

A  connecting  rod  (Fig.  31)  is  well  adapted  for  this  experi- 
ment. The  mass  M  of  the  rod  is  measured  once  for  all  and  given 
to  the  student  by  the  instructor.  The  center  of  gravity  is  located 
upon  a  line  C  which  is  found  by  balancing  the  rod  upon  a  knife 
edge.  This  line  should  be  carefully  marked  on  the  bar. 


Fig.  31. 

If  the  rod  is  now  hung  from  a  knife  edge,  placed  inside  the 
circular  hole  at  either  end,  it  forms  a  physical  pendulum  whose 
time  of  vibration  is  given  by  the  equation 


T=2^ww  (72) 

where  /  is  the  moment  of  inertia  about  the  point  .of  suspension 
and  h  the  distance  from  the  point  of  suspension  to  the  center  of 
gravity  of  the  rod.1 

Let  the  rod  be  swung  first  about  the  point  A  whose  distance  h± 
from  the  line  C  must  be  accurately  measured.  Determine  the 
period  7\  by  the  method  of  article  44.  Then 


r' 


where  /±  is  the  moment  of  inertia  about  the  point  A.    From  this 
equation  I±  is  calculated. 

Next,  suspend  the  rod  from  point  B  whose  distance  from  C 
is  h.2.    If  the  time  of  vibration  in  this  case  be  T2  we  have 


V 


where  I2  is  the  moment  of  inertia  about  point  B  and  is  calculated 
from  the  last  equation. 

1  College  Physics,  Article  55. 


MOMENT  OF  INERTIA  75 

The  moment  of  inertia  about  the  center  of  gravity  is2 

h  =  h  —  Mh\  (73)- 

Similarly 


Calculate  70  from  both  equations  and  take  the  mean  as  the  final 
result. 

FORM  OF  RECORD. 

Exercise  25.     To  find  the  moment  of  inertia  of  a  connecting 
rod  about  its  center  of  gra-vity. 


M=  ......  hi  =  ...... 

9  =  ......  hi  =  ...... 

Determination  of  Ti  Determination  of  T* 

(Record  as  in  article  44) 
/i  —  ......  /a  =  ...... 

Io=  ......  Io=  ...... 

Mean  h  =  ...... 

64.  Exercise  26.  Moment  of  Inertia  from  Torsional  Vi- 
brations. It  is  frequently  of  importance  to  determine  the  moment 
of  inertia  of  a  body  whose  form  is  irregular  or  such  as  to  render 
its  determination  by  computation  difficult  or  impossible.  In  such 
case  the  moment  of  inertia  may  be  determined  by  the  method  of 
torsional  vibrations. 

Suspend  the  body  by  a  stout  wire  so  as  to  swing  freely  about  a 
vertical  axis,  with  its  surface  horizontal.  If  twisted  through  an 
angle  it  will  tend  to  return  to  its  position  of  equilibrium,  and  in 
doing  so  will  execute  simple  harmonic  vibrations  of  period  T.  If 
I  be  the  moment  of  inertia  of  the  body  and  J^  the  moment  of  the 
restoring  couple  per  unit  angular  twist,  then  from  equation 
(60)  we  have 


If  now  there  be  added  to  the  body  a  ring  whose  moment  of  inertia 


2  College  Physics,  Article  51 ;  Eq.  134. 


76  PHYSICAL    MEASUREMENTS 

IT,  may  be  readily  calculated  from  its  dimensions,  (Table  VIII), 
then  the  period  of  the  system  becomes 


(74) 


Eliminating  ^"from  these  two  equations  we  have 


(75) 


Again  if  to  the  original  system  there  be  added  a  pair  of  cylin- 
ders, each  of  mass  M,  and  radius  r,  symmetrically  placed,  each  at 
distance  a,  from  the  axis  of  rotation,  the  period  of  the  combined 
system  becomes 


(76) 


where  /c,  the  moment  of  inertia  of  a  single  cylinder  about  the  axis 
of  rotation,  is  given  by  the  equation1 

Mr3 
/e  =  — -  +Ma2  (77) 


Combining  equations  (60)  and  (76)  we  find 

/  =  2/o.       r^r  (78) 

Finally,  if  to  the  original  system  we  add  both  ring  and  cylinders 
and  determine  the  period  of  vibration  T3f  of  the  system,  we  secure 
a  third  value  for  /  in  the  same  way  as  above,  from  the  equation 


.  nr  (79) 

The  body  to  be  investigated  may  have  the  form  of  a  rectangular 


1  College  Physics,  Article  51. 


MOMENT  OF  INERTIA  77 

bar  as  the  magnet  of  a  magnetometer,  or  the  form  of  a  circular 
cylinder  or  wheel,  or  it  may  be  of  any  irregular  outline,  so  long  as 
it  be  furnished  with  a  stout  hook  or  other  device  for  suspending  it 
from  an  axis  that  shall  pass  vertically  through  its  center  of  grav- 
ity. Its  surface  when  suspended  should  be  horizontal,  and  should 
have  marked  upon  it  lines  to  insure  the  exact  placing  of  the  ring 
and  cylinders.  The  position  of  the  centers  of  the  cylinders  should 
be  clearly  indicated  in  order  to  facilitate  the  determination  of  a. 
A  line  connecting  these  centers  must  pass  through  the  axis  of 
rotation  of  the  body. 

The  ring  should  be  of  rectangular  cross  section,  with  sides  and 
edges  well  polished,  and  a  diameter  clearly  marked  upon  each. 
The  cylinders  should  be  accurately  turned,  as  nearly  alike  as  pos- 
sible, and  have  sufficient  mass  to  insure  a  distinct  difference  in  the 
period  of  vibration  when  they  are  added.  The  ring  should  have 
a  mass  of  at  least  one  kilogram. 

The  exercise  comprises  the  following  measurements : 

(a)  Measure  the  external  and  internal  diameters  of  the  large 
ring  by  means  of  the  vernier  caliper  and  compute  the  external 
and  internal  radii,  1\  and  r2. 

(b)  Measure  on  the  dividing  engine  the  distance  between  the 
centers  of  the  two  cylinders  and  compute  the  length  a,  from  the 
axis  of  rotation  of  the  system  to  the  center  of  either  cylinder  when 
placed  upon  the  body. 

(c)  Measure  the  diameter  of  the  cylinders  and  compute  the 
mean  radius  r. 

(d)  Determine  the  mass  of  large  ring  MT,  and  of  either  of 
the  two  cylinders  Afc. 

(e)  Determine  to  thousandths  of  a  second  the  time  of  vibra- 
tion of  the  body  alone,  T ;  of  the  body  and  ring,  7\ ;  of  the  body 
and  cylinders,  T2 ;  of  the  body,  ring  and  cylinders,  T3. 


78 


PHYSICAL  MEASUREMENTS 


FORM  OF  RECORD. 


Exercise  26.    To  determine  the  moment  of  inertia  of  a  body  by 
torsional  vibrations. 

(a)  Enter  results  as  in  Art.  44. 


(k)  Value 

Mr 
?r 

Me 
a 

Ic 

Value 

r 

T\ 
T, 

2Y+T 

Computation  : 

colog  ( 
colog  ( 

2logMrL 

COlog  2  i= 

Ti—T 

log  /r  = 

logT'  = 

1  = 

65.  Exercise  27.  The  Ballistic  Pendulum.  When  two  bod- 
ies collide  in  such  a  way  that  their  velocities  at  the  instant  of  col- 
lision lie  along  the  common  normal  at  the  point  of  contact  we 
know  from  Newton's  Third  Law  of  Motion  that  the  sum  of  the 
momenta  of  the  two  bodies  is  unchanged  by  the  collision.  If  m± 
and  m2  be  the  masses  of  the  two  bodies,  z\  and  v2,  v\  and  v\  their 
respective  velocities  before  and  after  collision,  then 


mi  Vi  +  W2  Va  = 


(80) 


Newton  also  showed  that  under  the  foregoing  conditions  the 
relative  velocities,  before  and  after  impact,  bore  a  constant  ratio 
to  each  other,  or  if  we  assume  v±  to  be  the  velocity  before  impact 
of  the  more  rapidly  moving  body,  we  have 

1/1  —  v\  —  e  (vi  —  %)  (81  ) 

where  e  is  a  quantity  depending  upon  the  material  of  which  the 
bodies  are  composed.  Investigations  by  Hodgkinson,1  and  by 
Vincent2  show  that  e  is  influenced  in  some  degree  by  the  initial 
velocity.  The  quantity  e  is  termed  the  coefficient  of  restitution, 
and  is  always  less  than  unity. 

In  the  laboratory  the  foregoing  principles  are  easily  verified  by 
the  study  of  tne  collisions  between  suspended  spheres,  where  the 
direction  of  motion  may  be  controlled  and  the  respective  velocities 

1  Hodgkinson,  Report  of  British  Association,  1834. 

2  Vincent,  Proc.  Cambridge  Phil.  Society,  Vol.  X,  p.  332. 


MOMENT  OF  INERTIA 


79    - 


may  be  readily  measured.  For  convenience  of  adjustment  the 
spheres  are  taken  of  the  same  size,  and  the  masses  varied  by  mak- 
ing them  of  different  material. 

The  apparatus  (Fig.  32),  consists  of 
two  spheres  of  wood,  ivory,  iron  or 
lead,  about  9  cms  in  diameter,  hung 
on  bifilar  suspensions,  about  250  cms 
in  length,  and  so  adjusted  as  to  swing 
accurately  parallel  to  the  graduated 
arc  ssf.  The  spheres  carry  at  the  lower 
ends  of  their  vertical  diameters  small 
pointers  by  means  of  which  their  posi- 
tions may  be  read  off  on  the  graduated 
arc.  When  at  rest  the  spheres  must 
touch  lightly,  and  their  centers  lie  on 
an  arc  parallel  to  the  scale. 

Read  the  position  of  the  pointers, 
and  displace  one  of  the  spheres,  say  the  lighter  one,  through  an 
arc  of  not  more  than  20°.  Read  off  the  displacement  BC  =  c 
accurately  from  the  graduated  scale,  and  allow  the  sphere  to 
swing  back  against  the  one  at  rest.  Determine  as  accurately  as 
possible  the  extreme  positions  of  the  two  spheres  after  impact, 
taking  the  mean  of  at  least  five  simultaneous  readings  for  each. 
Two  observers  are  required. 

Repeat  the  experiment  by  allowing  the  heavier  sphere,  raised 
through  a  smaller  arc  to  swing  against  the  lighter  one  and  taking 
readings  as  before.  In  all  cases  take  care  to  avoid  giving  a  spin- 
ning motion  to  the  released  sphere.  The  masses  of  the  spheres 
are  to  be  determined  by  the  student  unless  furnished  by  the  in- 
structor in  charge. 

From  Fig.  32  it  is  seen  that1 


Fig.  32. 


and  also 
whence 


?  =  2g  A  B 


2OB 


(82) 

(83) 

(84) 


1  College  Physics,  Article  25. 


8o 


PHYSICAL,    MEASUREMENTS 


and  the  velocity  vlt  of  the  impinging  sphere  is  directly 'propor- 
tional to  the  chord  C  B  =  c,  of  the  arc  through  which  it  swings. 
For  angles  not  exceeding  20°  the  arc  may  be  set  equal  to  the 
chord  to  within  one-third  of  one  per  cent.  We  may  therefore 
insert  readings  from  the  graduated  arc  in  equation  (80)  instead 
of  the  actual  velocities. 

Apply  equation  (80),  in  which  v2  is  to  be  taken  equal  to  zero, 
and  vlf  the  velocity  of  the  impinging  sphere,  is  to  be  regarded 
as  positive  in  each  case,  and  the  reverse  direction  as  negative. 

FORM  OF  RECORD. 

Exercise  27.  The  ballistic  pendulum.  To  prove  the  law  of 
conservation  of  momentum. 

Date. . 


Sphere 

1  .Material 

2  Material 

No. 


z/»  —  v\ 


CHAPTER  IV. 

DENSITY. 

66.  Definition.    Density  is  denned  as  mass  per  unit  volume, 
and  is  therefore,  in  the  c.  G.  s.  system,  measured  by  the  number 
of  grams   per  cubic   centimeter.     The   International   Bureau   of 
Weights  and  Measures  however,   for  practical  reasons,  chooses 
the  milliliter  as  the  unit  of  volume,  where  the  liter  is  denned  as 
the  volume  occupied  by  one  kilogram  of  distilled  water  at  4°C 
(=  1000.05  cm3).     The  density  of  water  at  this  temperature  be- 
comes then  strictly  equal  to  unity,  or  one  gram  per  milliliter. 

According  to  this  definition  of  unit  volume,  the  numerical  value 
for  the  density  of  any  substance  becomes  identical  with  that  denot- 
ing its  specific  gravity.  In  what  follows  no  distinction  will  be 
made  between  the  two  definitions,  since  the  degree  of  accuracy 
attainable  by  the  methods  described  below,  does  not  warrant  such 
a  distinction. 

67.  Exercise  28.    Density  from  Mass  and  Volume. 

FORM  OF  RECORD. 

Exercise  28.  To  determine  the  density  of  a  brass  cylinder  from 
its  volume  and  mass. 


Density  of  brass  cylinder  No Date.. 

Length  Diameter 

Mass, 


Mean Mean 

Volume Density. 


68.     Exercise  29.     The  Pyknometer.    The  pyknometer  in  its 

simplest  form  consists  of  a  glass  vessel  (Fig.  33),  provided  with 


82  PHYSICAL    MEASUREMENTS 

an  accurately  ground  stopper,  perforated  through- 
out its  length.  Before  use  it  is  to  be  thoroughly 
cleaned  and  dried  with  alcohol  or  ether.  It  is 
then  carefully  weighed.  Call  this  weight  W^. 
The  pyknometer  is  then  rilled  with  distilled  water, 
placed  in  a  bath  of  water  at  30°  C.  and  allowed  to 
remain  there  five  minutes.  (Why?)  It  is  then 
wiped  dry,  and  its  weight,  Wlf  determined.  After  Fig.  33. 

cleaning  and  drying  as  before,  the  pyknometer  is  filled  with  the 
liquid  under  examination,  brought  to  the  temperature  of  30°  C, 
wiped  dry  and  weighed.  Call  this  weight  W2.  Derive  and  apply 
formula  for  the  density  of  a  liquid.  Correct  for  temperature,  by 
multiplying  dQ,  the  value  found,  by  dlf  the  density  of  the  standard1 
(for  water  at  3O°C,  d1  =  0.9958  g  per  cc).  Unless  the  values  are 
to  be  carried  to  more  than  three  decimal  places,  the  buoyant  force 
of  the  air  may  be  neglected.  Instead  of  making  all  the  weighings 
at  30°,  any  other  temperature  may  be  taken  provided  it  be  higher 
than  the  temperature  of  the  room  where  weighings  are  made 
(Why?) 

FORM  OF  RECORD. 

Exercise  29.    Determine  the  density  of  two  liquids. 

Pyknometer  No Date 

Density  of 


Corrected  density,  d,  of = 

69.  Exercise  30.  Mohr's  Balance.  In 
Mohr's  balance  (Fig.  34),  the  beam  is  di- 
vided into  ten  equal  parts  of  which  the  last 
coincides  with  the  end.  Upon  this  end  is 
hung  by  means  of  a  fine  platinum  wire  a 
small  sinker  containing  a  thermometer. 
The  weights  consist  of  four  pairs  of  riders 
weighing  respectively  m,  o.i  m,  o.oi  m, 
o.ooi  m  grams.  The  instrument  is  usually 
so  adjusted  that  the  sinker  is  exactly  coun- 
terpoised in  air.  When  immersed  in  water 

1  College  Physics,  Article  69. 
Also  Stewart  and  Gee,  Practical  Physics,  Vol.  I,  p.  119. 


DENSITY  83 

at  4°  C  the  buoyant  force  on  the  sinker  is  equal  to  the  weight  of 
the  largest  rider.  When  the  sinker  is  put  into  any  other  liquid 
the  weights  needed  to  equal  the  buoyant  force  upon  the  sinker 
in  each  case  are  in  direct  proportion  to  the  densities  of  these 
liquids.  If  in  be  called  unity  then  the  density  of  the  liquid  can 
be  read  directly  from  the  beam.  The  balance  must  show  for 
water  at  t°  C  the  density  corresponding  to  this  temperature  as 
given  in  Table  II.  If,  instead  of  this  density  d,  it  show  a  density 
dlf  then  all  results  must  be  multiplied  by  d/d±.  It  is  obvious  that 
the  instrument  may  be  used  as  an  ordinary  balance  as  well,  and 
the  densities  of  solids  and  liquids  referred  to  water  at  any  tem- 
perature may  be  determined  by  means  of  it  with  equal  ease. 

FORM  OF  RECORD. 

Exercise  jo.     Mohr's  balance.     Redetermine  the  densities  of 
the  substances  used  in  Exercises  28  and  29. 


i.     For  solids:  Date  ......  , 

Density    of  .................. 

Weight  in  air  ............... 

Weight  in   water  ............  Temperature  of  water. 


Corrected    density 


For  liquids  :  Date. 

Density    of 

Temperature    

Buoyancy  of  sinker  in   water 

Buoyancy  of  sinker  in  liquid 

Density.  . 


CHAPTER  V. 

SURFACE  TENSION  AND  VISCOSITY. 

70.  Characteristics  of  a  Liquid.     A  liquid  is  a  body  which 
has  no  elasticity  of  shape,  or  which  yields  continuously  under 
the  action  of  a  shearing  stress.    It  is  characterized  by  consider- 
able mobility  of  its  molecules,  by  a  distinct,  free  upper  surface, 
usually  of  the  meniscus  shape  when  confined  in  a  tube,  and  by 
the  existence  in  that  free  surface  of  a  specific  stress  or  tension 
not  found  elsewhere  in  the  body.    As  a  result  of  this  tension  the 
liquid  behaves  as  if  it  were  enclosed  in  an  elastic  bag  which  tends 
to  contract  indefinitely  and  compress  the  liquid  into  as  small  a 
volume  as  possible. 

If  the  liquid  exist  in  the  form  of  a  film  then  the  two  sides  of 
the  film  exhibit  this  tension  in  like  degree  and  the  film  tends  to 
contract  indefinitely  unless  prevented  by  the  application  of  ex- 
ternal force.  If  the  external  force  required  to  keep  a  film,  /  units 
in  width,  in  equilibrium  be  F  dynes,  then  71,  the  surface  tension 
of  the  film,  measured  in  dynes  per  centimeter,  is  given  by  the 
equation1 

2Tl  =  F  (85) 

or 

T=  7T  (86) 

As  the  surface  tension  of  any  liquid  is  greatly  modified  by  the 
presence  of  any  substance  dissolved  in  it,  it  is  of  the  highest 
importance  in  experiments  on  surface  tension  that  all  surfaces 
coming  in  contact  with  the  liquid  be  absolutely  clean,  and  that  the 
liquid  itself  be  pure. 

71.  Exercise  31.     Measurement  of  Surface  Tension.     The 
apparatus  consists  of  a  Jolly  balance  and  several  small  frames  of 

1  College  Physics,  Article  89. 


SURFACE  TENSION 


different  width  made  from  short  pieces  of  fine  glass  tubing  or  fine 
platinum  wire,  with  the  ends  bent  so  as  to  form  three  sides  of  a 
rectangle,  all  lying  in  the  same  plane.  Suspend  one  of  these 
frames  from  the  spring  of  a  Jolly  balance  by  a  fine  thread  or  wire 
after  removing  the  pan,  and  adjust  so  that  the  sides  of  the  frame 
hang  vertically  and  the  ends  just  dip  into  the  liquid.  Take  the 
reading  on  the  Joly  balance,  making  at  least  three  adjustments 
and  taking  the  mean.  Next  form  the  film  by  raising  the  vessel 
containing  the  liquid  until  the  horizontal  side  of  the  frame  is 
immersed  and  lowering  the  vessel  again  to  its  former  position. 
The  tension  in  the  film  tends  to  pull  the  frame  back  into  the  liquid 
and  elongates  the  spring.  Raise  the  frame  by  the  rack  and  pinion 
movement  until  the  indicator  has  reached  its  former  zero  position. 
Note  the  reading.  Make  four  determinations  with  different 
lengths  of  the  film  for  each  frame,  using  frames  of  three  different 
widths. 

Determine  the  constant  of  the  spring  of  the  balance  by  placing 
gram  masses  upon  the  pan  and  noting  the  elongation  per  gram. 
Calculate  the  force  m  g,  in  dynes  exerted  by  the  spring  of  the  bal- 
ance to  hold  the  film  in  equilibrium.  The  surface  tension  of  the 
film  in  dynes  per  centimeter  width,  is  given  by 

T=  ^  (87) 


FORM  OF  RECORD. 

Exercise  ji.     To  determine  the  surface  tension  of  water,  and 
of  a  soap  solution  at  room  temperature. 


Constant  o 
Mass 

f  Jolly  Balance 
Reading 

Width  of  Frame 

i 

2 

3 

4 

Zero 

Re 

Water 

a  dings 
Soap 
Solution 

Force  in  dynes  to  stretch  spring  one  scale  division,  =  .  . . . 
Surface  Tension 


Frame  No. 


Water 


Mean 


Soap  Solution 


86  PHYSICAL    MEASUREMENTS 

72.  Exercise  32.  Surface  Tension  from  Capillary  Action. 
If  a  freshly  drawn  glass  tube  of  small  diameter  be  held  vertically 
and  lowered  into  clean  water,  the  surface  action  of  the  water 
drags  the  liquid  up  the  tube  to  a  height  h,  at  which  the  weight  of 
the  liquid  column  just  balances  the  vertical  component  of  the 
force  due  to  surface  tension,  and  equilibrium  ensues.  If  r  be  the 
radius  of  the  tube  and  0  the  angle  of  contact  between  the  liquid 
and  the  tube  and  d  the  density  of  the  liquid,  then  for  equilibrium 
we  have 

2TrrTcos9  =  Trrhdg  +  vdg  (88) 

where  v  is  the  volume  of  the  liquid  forming  the  meniscus. 

If  the  liquid  wet  the  tube  6  may  be  set  equal  to  zero,  and  if  we 
assume  the  radius  to  be  so  small  that  the  cup  formed  by  the  men- 
iscus is  hemispherical,  then 


(89) 


and  the  expression  for  T  becomes 


or 


where  h'  is  the  distance  from  the  surface  of  the  liquid  to  a  point 
r/3  above  the  lowest  point  of  the  meniscus. 

Place  a  freshly  drawn  glass  tube,  of  less  than  one  millimeter 
diameter,  in  a  glass  cup  with  straight  sides  and  pour  in  two  or 
three  centimeters  depth  of  distilled  water.  Move  the  tube  up  and 
down  through  a  range  of  a  centimeter  or  two  so  as  to  make  sure 
that  the  meniscus  is  formed  against  a  wet  surface.  Measure  with 
a  finely  divided  scale  or  better  with  a  micrometer  cathetometer, 
the  height  h'.  Mark  the  point  on  the  tube  reached  by  the  water 
column  and  with  a  fine  file  cut  the  tube  at  this  point.  Measure 
the  diameter  of  the  tube  with  the  micrometer  microscope,  taking 
the  mean  of  at  least  three  readings.  Note  the  temperature  and 
insert  in  the  formula  the  appropriate  value  for  d.  Compute  the 
value  of  the  surface  tension  for  pure  water  at  the  observed  tern- 


VISCOSITY  87 

perature.    Repeat  the  experiment  with  a  dilute  solution  of  fuch- 
sine  in  alcohol. 

FORM  OF  RECORD. 

Exercise  32.  To  determine  the  surface  tension  of  distilled 
zuater  and  of  an  alcoholic  solution  of  fuchsine  at  room  tempera- 
ture, 

Temperature  ....................  Date  ................ 

Readings  Diameter  of  tube 

Surface  of  liquid        Meniscus         I 


T  for  water  =  ......  T  for  fuchsine  solution  =  ...... 

73.  Coefficient  of  Viscosity.  As  has  been  shown  in  a  pre- 
vious article,  a  liquid  yields  continuously  under  the  action  of  a 
shearing  force.  An  ideal  liquid  would  offer  no  resistance  to  such 
a  force  but  would  suffer  its  molecules  to  glide  past  each  other 
without  any  loss  of  energy.  Most  liquids  however  offer  some 
resistance  to  a  shearing  force  and  this  property  by  virtue  of  which 
a  liquid  resists  the  relative  motion  of  its  parts  is  termed  viscosity. 

The  coefficient  of  viscosity  is  defined  as  the  constant  ratio  of 
the  shearing  stress  in  a  fluid  to  its  time  rate  of  change  of  shear- 
ing strain.  The  latter  is  given  by  l/Lt,  equal  to  v/L.  If  F 
be  the  force  acting  on  one  of  two  parallel  surfaces  A  of  a  fluid, 
separated  by  a  distance  L  cm  and  moving  with  a  velocity  v  rela- 
tive to  each  other,  the  coefficient  of  viscosity  of  the  fluid  is 


L 

The  coefficient  of  viscosity  is  therefore  numerically  equal  to 
"the  tangential  force  on  unit  area  of  either  of  two  horizontal 
planes  at  unit  distance  apart,  one  of  which  is  fixed,  while  the  oth- 
er moves  with  the  unit  of  velocity,  the  space  between  being  filled 
with  the  viscous  substance."1 

This  quantity  may  be  measured  either  by  determining  the  flow 
of  a  viscous  fluid  through  a  tube  of  small  bore,  or  by  determin- 
ing the  resistance  offered  by  the  fluid  to  the  passage  of  a  solid 
body  through  it. 


1  Maxwell,  Theory  of  Heat. 


88 


PHYSICAL    MEASUREMENTS 


74.  Exercise  33.  Coefficient  of  Viscosity  by  Flow  through 
a  Capillary  Tube.  It  is  shown  in  works  on  physics1  that  the  vol- 
ume of  a  liquid  discharged  in  time  t,  through  a  capillary  tube  of 
radius  r,  and  length  /,  under  a  pressure  of  p  dynes  per  square  cen- 
timeter, is  given  by  the  equation 


8r)l 


(93) 


where  V  is  the  volume,  d  the  density,  77  the  coefficient  of  viscosity 
of  the  liqui'd,  and  h  the  height  from  which  it  flows.     From  this 

it  follows  that 

?r/>r4  ifhdg  r4    ±        *  h  d3  g  r* 


v  i 


-t, 


(94) 


when  m  is  the  mass  of  the  liquid  discharged  through  the  tube. 


To  determine  this  coeffi- 
cient in  absolute  meas- 
ure, each  of  the  above 
quantities  must  be  ex- 
pressed in  absolute  units. 
Select  a  capillary  tube 
of  uniform  bore  and 
clean  it  thoroughly  by 
running  through  it  re- 
peatedly a  solution  of 
sodium  bi-chromate  and 
strong  sulphuric  acid, 
and  afterward  washing 
it  well  with  distilled 
water.  Fasten  the  tube, 
by  means  of  a  rubber 
stopper,  to  the  lower  end 
of  a  large  separating 
flask,  as  shown  in  Fig.  35. 


Fig.  35- 


Fig.  36. 


Pour  in  some  of  the  liquid  to  be  studied  and  allow  it  to  flow 
through  the  capillary  for  a  short  time.     Now  close  the  stopcock, 


1  Poynting  and  Thompson,  Properties  of  Matter,  Chapter  XVIII. 


VISCOSITY  8c> 

fill  the  flask  with  the  liquid  under  examination,  and  arrange  for 
constant  head  during  the  flow,  by  inserting  in  the  upper  end  of 
the  flask  a  stopper  through  which  passes  a  piece  of  glass  tube  of 
wider  bore,  reaching  well  down  into  the  liquid  as  shown  in  the 
figure. 

Place  under  the  lower  end  of  the  capillary  a  wide  beaker,  or 
flat  dish,  and  pour  in  sufficient  liquid  to  cover  the  end  of  the  capil- 
lary tube.  Weigh  this  vessel  with  its  contents.  Replace  the  beak- 
er, open  the  stopcock  and  allow  the  liquid  to  flow  for  a  definite 
time  t.  Weigh  the  beaker  and  its  contents  a  second  time  and 
determine  m,  the  mass  of  liquid  discharged.  Determine  the  dens- 
ity of  the  liquid  if  necessary  by  means  of  Mohr's  balance,  the 
lengths  /  and  h,  by  means  of  a  meter  stick  and  the  radius  of  the 
tube  by  filling  a  measured  length  of  it  with  mercury,  weighing  it, 
and  computing  the  average  radius  from  the  length,  mass,  and 
density  of  the  mercury.  Use  three  capillary  tubes  of  different 
radii. 

For  a  determination  of  the  specific  viscosity,  i.  e.,  the  ratio 
between  the  viscosity  of  the  liquid  and  that' of  water  at  o°C,  or  at 
room  temperature,  the  dimensions  of  the  capillary  tube  need  not 
be  known.  By  performing  the  experiment,  with  the  apparatus 
unchanged,  first  with  water  and  afterward  with  the  liquid,  we* 
have 

—  =  j™***  .  (95) 

•n  2       #2  iMifi 

or 

^=4^  (96) 


if  t,  the  time  of  flow  be  made  the  same  in  each  case. 

The  exercise  may  be  varied  by  allowing  equal  volumes  of  the 
two  liquids  to  flow  through  the  capillary,  and  comparing  the  times 
required,  from  which  we  have 

—  =  TT.  (97) 

77  2        d~t~ 

In  this  case  a  tube  as  shown  in  Fig.  36  is  very  convenient. 
Draw  the  liquid  into  the  tube  until  it  rises  above  the  upper  mark.. 


PHYSICAL    MEASUREMENTS 


Then  let  it  run  out  and  note  the  time  it  takes  from  the  instant  it 
passes  this  mark  until  the  liquid  enters  the  capillary.  The  time 
may  be  determined  conveniently  by  means  of  a  stop  watch. 

FORM  OF  RECORD. 

Exercise  jj.   To  'determine  the  coefficient  of  viscosity  of  water 
at  room  temperature. 

Temperature  of  water 

No.  of  tube 

Length 

Mass  of  mercury 

Volume  of  tube 

Average   radius 


Date 

2 

3 

No.  of  tube 
Time  of  flow 
Mass  of  beaker  after 
Mass  of  beaker  before 
Mass  of  water 

i 

2 

3 

Mean  = 


CHAPTER  VI. 

MEASUREMENTS  IN  SOUND. 

75.  Exercise  34.  Velocity  of  Sound  in  Metals.  (Kundt's 
Method.)  A  brass  rod  held  firmly  clamped  at  its  middle  point 
when  stroked  with  a  resined  cloth  vibrates  longitudinally  like  the 
air  in  an  open  organ  pipe  when  sounding  its  fundamental  tone. 
The  middle  of  the  rod  being  rigidly  fixed  is  obviously  a  node  and 
the  length  of  the  rod  is  therefore  the  half  wave-length  in  brass 
of  the  sound  produced.  If  the  end  of  the  rod  be  brought  into  con- 
tact with  an  enclosed  column  of  air  whose  length  may  be  varied 
at  will,  it  is  possible  so  to  adjust  the  length  of  the  air  column  as 
to  render  it  capable  of  vibrating  in  unison  with  the  rod.  In  this 
case  the  enclosed  air  column  having  been  thrown  into  stationary 
vibration  behaves  as  a  resonator  closed  at  both  ends;  it  must 
therefore  contain  at  least  one,  and  usually  contains  a  number  of 
half  wave-lengths,  of  the  sound  in  air,  produced  by  the  rod. 
From  the  fundamental  equation  connecting  velocity,  frequency 
and  wave-length,  we  have1 


where  N  is  the  frequency  and  V  ,  v,  A,  and  A'  denotes  the  velocities 
and  wave-lengths  of  the  same  sound  in  brass  and  in  air  respect- 
ively. From  this  we  have  at  once 

r=sv.  £7  (99) 

where  v,  the  velocity  of  sound  in  air  must  be  corrected  for  tem- 
perature according  to  the  formula2 


rt  =  vQ  V  ( i  -\-  at)  ( 100) 

where  a  for  air  at  ordinary  humidity,  may  be  taken  as  0.004  Per 
degree  C. 


1  College  Physics,  Article  104. 

2  College  Physics.  Article  114. 


92  PHYSICAL    MEASUREMENTS 

In  practice  a  brass  rod  about  one  centimeter  in  diameter  and 
one  meter  long  is  clamped  in  a  vise  at  its  middle  point  and  bears 
at  one  end  a  small  disk  of  paper.  A  glass  tube  about  5  cms  in 
diameter  and  150  cms  long  (Fig.  37),  has  one  end  closed  air- 
tight by  a  sheet  of  rubber  membrane  tied  smoothly  over  the  end, 
while  the  other  end  is  furnished  with  an  adjustable  piston  sliding 
freely  in  the  tube.  The  walls  of  the  tube  are  lightly  dusted 
throughout  with  fine  cork  filings  or  amorphous  silica.  The  tube 
is  placed  horizontally  upon  two  V-shaped  wooden  supports  so  that 
the  rubber  membrane  presses  lightly  against  the  disk  of  stiff  paper 
on  the  end  of  the  rod.  The  rod  is  set  in  vibration  by  chafing  it 
gently  with  a  piece  of  cloth  or  chamois  skin  covered  with  pow- 
dered resin.  The  cloth  or  chamois  skin  should  be  held  between 
the  thumb  and  fore  finger  of  each  hand  and  pressed  lightly  against 


Fig-  37- 

each  side  of  the  rod.  When  the  rod  is  properly  clamped  a  very 
slight  pressure  is  sufficient  to  produce  a  loud,  clear  tone.  Adjust 
the  piston  in  the  outer  end  of  the  tube  until  the  enclosed  air 
vibrates  freely  on  stroking  the  rod.  This  is  indicated  by  the 
powder  being  tossed  about  in  the  tube  and  falling  in  the  charac- 
teristic figures  shown  above.  The  nodes  are  indicated  by  small 
rings  of  powder  and  the  antinodes  by  transverse  layers  or  striae. 
The  value  of  A'/2  is  found  by  measuring  over  a  number  of  circles 
from  center  to  center,  and  dividing  the  distance  by  the  number  of 
spaces  or  loops  measured,  remembering  that  the  distance  from 
node  to  node  is  equal  to  A'/2.  It  will  be  noticed  that  a  node  is 
found  both  at  the  rubber  diaphragm  and  at  the  piston.  Why? 

Measure  over  as  large  a  number  of  loops  as  possible  and  com- 
pute A'/2.  Tap  the  tube  lightly,  rolling  it  over  and  over  until  the 
powder  is  evenly  distributed,  and  repeat  the  determination.  Take 
at  least  five  separate  sets  of  measurements.  Avoid  heating  the 
rod  by  undue  pressure  or  by  continued  rubbing. 


EXPERIMENTS   IN    SOUND 


93 


FORM   OF   RECORD. 

Exercise  34.    To  determine  the  velocity  of  sound  in  brass. 


Temperature         

Length  of 

rod  

Date  

Number  of  loops 

Distance 

A' 

X 

between  nodes 

2 

2 

\ 

\'         



v  —  

Velocity  of  sound  in  brass  — 


Mean. 


76.     Exercise  35.    Computation  of  Young's  Modulus.    From 
the  equation  for  the  velocity  of  sound  in  any  medium 


v= .  -4- 


where  e  is  the  coefficient  of  elasticity,  and  d  the  density  of  the 
medium  in  question,  we  may  compute  e  at  once  from  the  data 
obtained  in  previous  experiments.  In  the  case  of  longitudinal 
waves  transmitted  through  solids  the  coefficient  of  elasticity  in- 
volved is  M,  Young's  modulus,  whence 


M  =  V*  d  . 


FORM    OF   RECORD. 


(102) 


Exercise  55.    To  compute   Young's  modulus  for  brass  from 
velocity  of  sound  and  density. 


Date 

V  as  found  in  Exercise  34  = 
d  as  found  in  Exercise  28  = 


Compare  result  with  that  obtained  in  Exercises  18  and  20. 

77.  Exercise  36.  Rating  a  Tuning  Fork.  Graphical  Meth- 
od. A  tuning  fork,  one  prong  of  which  is  armed  with  a  fine 
sharp  tip  of  flexible  sheet  copper,  is  mounted  at  right  angles  to  a 


94  PHYSICAL    MEASUREMENTS 

metallic  cylinder.  The  cylinder  is  carried  upon  an  axis  furnished 
with  a  thread  so  that  when  rotated  it  is  advanced  longitudinally 
at  the  same  time,  so  that  the  tracing  point  generates  a  spiral  upon 
the  surface  of  the  cylinder.  If  the  surface  of  the  cylinder  be 
slightly  smoked,  the  tuning  fork  set  in  vibration  and  the  cylinder 
rotated  uniformly  the  tracing  point  describes  a  sinusoidal  curve 
upon  its  surface.  The  number  of  vibrations  executed  in  a  sec- 
ond being  thus  automatically  recorded  by  the  fork,  it  is  only 
necessary  to  indicate  the  beginning  of  the  successive  seconds 
upon  the  curve  in  order  to  read  the  vibration  frequency  of  the 
fork  directly  from  the  surface  of  the  cylinder. 

In  practice  the  cylinder  is  covered  with  a  sheet  of  firm  smooth 
paper  pasted  smoothly  on  by  gumming  one  end  of  the  paper  and 
pressing  the  gummed  surface  upon  the  other.  The  paper  must 
not  be  stuck  upon  the  cylinder.  The  paper  is  then  smoked  uni- 


Fig.  38. 

formly  and  lightly  by  means  of  a  gas  flame  passed  back  and  forth 
near  the  paper  while  the  cylinder  is  continuously  rotated,  allowing 
only  about  an  inch  of  the  tip  of  the  flame  to  touch  the  paper. 

Care  should  be  taken  not  to  smoke  the  paper  too  black.  The 
fork  is  then  adjusted  in  its  holder  so  that  the  point  just  touches 
the  paper  at  the  highest  part  of  the  cylinder  and  at  the  left  end  of 
the  cylinder  so  that  when  the  latter  is  rotated  the  fork  seems  to 
move  from  left  to  right  along  the  cylinder.  The  cylinder  must 
rotate  from  the  tracing  point.  The  time  intervals  are  recorded 
upon  the  paper  by  connecting  the  cylinder  and  the  fork  to  the 
secondary  terminals  of  an  induction  coil,  the  primary  circuit  of 
which  contains  a  suitable  battery  and  is  closed  by  a  pendulum 
beating  seconds.  Consequently  when  the  tracing  point  rests  upon 
the  paper  and  the  coil  is  put  in  action  a  spark  passes  from  the 
point  to  the  cylinder  each  time  the  circuit  is  closed,  i.  e.,  every 
second.  The  passage  of  this  spark  leaves  a  small  spot  on  the 
smoked  surface  thus  marking  the  time  very  accurately.  The  ap- 
pearance of  the  sparks  should  be  like  that  shown  in  Fig.  38. 


EXPERIMENTS   IN    SOUND  95 

vSometimes  the  coil  will  give  two  or  three  sparks  instead  of  one. 
(Why?)  In  this  case  read  from  the  last  one.  The  fork  should 
be  so  adjusted  that  when  the  point  touches  the  paper  and  the  fork" 
is  properly  bowed  it  will  continue  to  vibrate  for  at  least  twelve 
seconds  before  coming  to  rest.  When  all  the  adjustments  are 
made  the  coil  is  put  in  action,  the  fork  bowed  and  the  cylinder 
rotated  uniformly  while  the  fork  continues  to  vibrate.  The  fork 
is  then  bowed  again  and  the  cylinder  rotated  as  before.  When  the 
paper  is  filled  the  fork  is  removed,  the  paper  cut  along  the  lapf 
parallel  to  the  axis  of  the  cylinder  almost  but  not  quite  apart ;  the 
cylinder  is  then  turned  over  and  the  paper  broken  loose.  In  this 
way  only  can  the  paper  be  removed  from  the  cylinder  without 
spoiling  the  record. 

The  paper  is  then  passed,  face  upwards,  through  a  fixing  solu- 
tion of  shellac  in  alcohol  and  then  dried.  In  a  few  minutes  the 
curve  is  ready  to  be  examined.  Count  the  waves  for  ten  seconds 
and  record.  In  counting  the  waves  always  take  an  even  number 
of  seconds. 

FORM   OF  RECORD. 

Hxercise  36.  To  determine  the  frequency  of  a  tuning  fork  by 
the  graphical  method. 

Date 

Seconds  Number  of  Vibrations :  :       N 


CHAPTER  VII. 

MEASUREMENTS  IN  HEAT. 

78.  Effects  of  Heat.     In  general,  all  physical  properties  of  a 
body,  except  its  mass  and  weight,  are  changed  by  adding  to  it 
heat  energy.     In  this  chapter  only  the  more  distinctive  of  such 
changes  will  be  considered,  among  which  the  following  may  be 
noted : 

(a)  The  temperature  of  the  body  rises. 

(b)  The  body  undergoes  a  change  in  volume;  in  general  an 
increase  in  volume  of  the  body  attends  an  increase  in  tempera- 
ture, if  the  pressure  remain  constant. 

(c)  The  body  exhibits  a  change  of  pressure,  if  its  volume 
be  kept  constant. 

(d)  The  body  may  change  its  state  or  condition,   as   for 
example,  ice  changes  to  water  and  water  to  steam  upon  the  addi- 
tion of  definite  quantities  of  heat. 

Other  changes,  such  as  changes  in  the  molecular,  optical, 
electrical  or  magnetic  properties  of  a  body,  incident  upon  the 
addition  of  heat  energy  to  it,  are  usually  investigated  only  in  so 
far  as  such  changes  are  dependent  upon  changes  of  temperature. 
The  percentage  variation  of  any  physical  property  of  a  body 
per  unit  change  of  temperature  is  termed  temperature  coeffi- 
cient, and  is  more  or  less  characteristic  of  the  substance  of 
which  the  body  is  composed.  The  determination  of  such  charac- 
teristic coefficients  falls  more  naturally  under  the  chapters  deal- 
ing with  those  properties. 

THERMOMETRY. 

79.  Thermometry.     If  two  bodies  possessing  different  tem- 
peratures be  brought  into  thermal  union,  heat  flows  from  the  one 


THERMOMETRY  97 

of  higher  temperature  to  the  one  of  lower  temperature,  and  in 
general  the  flow  of  heat  is  such  as  to  produce  and  maintain_an_ 
equilibrium  of  temperature  in  the  body,  or  system  of  bodies,  if 
thermally  insulated  from  all  other  bodies. 

Temperature  is  defined  quantitatively  in  terms  of  the  increase 
in  pressure  of  hydrogen  gas  of  constant  volume,  under  the  as- 
sumption that  equal  increments  of  temperature  produce  equal 
increments  of  pressure  in  the  gas.  As  points  of  reference  in  the 
measurement  of  temperature  the  two  fixed  points  for  water,  the 
freezing  and  the  boiling  points  under  standard  conditions,  have 
been  chosen.  In  the  centigrade  scale  the  temperature  of  melting 
ice  is  called  o°.  and  the  temperature  of  steam  forming  freely 
under  a  pressure  of  760  mm  of  mercury,  is  taken  as  100°. 

Temperatures  are  usually  measured  by  means  of  mercury-in- 
glass  thermometers.  Such  thermometers  possess  numerous  dis- 
advantages as  compared  with  the  gas  or  air  thermometer,  promi- 
nent among  which  are  the  following: 

(a)     Inequality  of  the  bore  of  the  glass  tube. 
(&)     Inequality  of  the  scale. 

(c)  Neither  glass  nor  mercury  expands  equally  and  uniform- 
ly throughout  any  large  range  of  temperature. 

(d)  The  instability  and  uncertainty  of  the  fixed  points  of  the 
thermometer,  arising  either  from  slow  changes  going  on  in  the 
thermometer  itself  or  from  sudden  and  large  variations  in  tem- 
perature incident  upon  the. use  of  the  thermometer. 

From  these  causes  it  is  evidently  a  matter  of  first  importance 
to  verify  the  readings  of  a  mercury-in-glass  thermometer,  with 
which  any  accurate  work  is  to  be  attempted. 

80.  Exercise  37.  Determination  of  the  Fixed  Points  of  a 
Thermometer.  The  fixed  points  of  an  ordinary  thermometer  are 
usually  in  error  by  some  fraction  of  a  degree  and  these  errors 
when  determined,  form  the  basis  of  correction  to  be  applied  to  all 
subsequent  readings  made  with  the  instrument.  The  fixed  points 


98 


PHYSICAL    MEASUREMENTS 


must  be  frequently  determined.    These  determinations  fall  under 
two  heads : 

(a)  The  Boiling  Point.  The  appara- 
tus (Fig.  39),  consists  of  a  brass  vessel 
partly  filled  with  water  and  having  in  its 
upper  part  double  walls  so  arranged  that 
the  steam  passes  up  through  the  inner 
cylinder,  down  through  the  outer  space 
and  escapes  from  a  short  tube  near  the 
bottom.  The  thermometer  is  passed  snug- 
ly through  a  close-fitting  cork,  into  the 
inner  cylinder.  The  bulb  is  not  allowed 
to  come  in  contact  with  the  water  and 
should  be  surrounded  by  wire  gauze  to 
prevent  overheating.  If  possible,  almost 
the  entire  filament  of  mercury  should  be 
enclosed  by  the  steam  issuing  freely  un- 
der atmospheric  pressure.  The  tempera- 
Fig.  39.  ture  of  the  steam  is  found  from  the  baro- 
metric pressure  under  which  the  water  boils,  by  means  of  the 
following  approximate  formula 


t  =  100°  +  0.0375  (B  —  760) , 


(103) 


where  B  denotes  the  barometric  pressure  in  mm,  corrected  to  o°C. 
For  accurate  values  of  boiling  point  at  different  pressures  see 
Table  IX. 

If  the  thermometer  reads  t  +  b,  instead  of  the  calculated  t  de- 
grees, the  correction  to  be  applied  to  this  reading  is  —  b  degrees. 

No  readings  should  be  taken  until  the  steam  issues  freely  from 
the  tube  at  the  bottom.  This  tube  should  be  kept  entirely  open 
and  the  water  should  not  be  boiled  too  violently.  (Why?) 

(b)  The  Zero  Point.  The  thermometer  is  now  removed  from 
the  boiling  point  apparatus  and  as  soon  as  the  temperature  has 
dropped  to  50°  is  plunged  into  a  clean  vessel  containing  a  mixture 
of  distilled  water  and  pure  ice  so  that  the  mercury  thread  is  en- 
tirely surrounded  by  the  mixture.  When  the  reading  is  taken 


THERMOMETRY  99 

the  thermometer  should  be  raised  just  high  enough  to  show  dis- 
tinctly the  upper  end  of  the  mercury  filament.  In  reading  care 
should  be  taken  to  avoid  parallax.  As  soon  as  the  mercury  has 
fallen  to  i°,  note  the  reading  every  minute  until  it  has  remained 
stationary  for  five  minutes.  Take  the  final  readings  as  the  zero 
reading  of  the  thermometer. 

What  would  be  the  correction  for  the  zero  point,  if  the  reading 
is  found  to  be  -}-  a  degrees  ? 

The  zero  point  thus  found  is  termed  the  depressed  zero,  since 
it  is  usually  lower  than  the  value  found  for  the  zero  point,  if  it  be 
determined  just  before  the  boiling  point  is  taken.  This  difference 
is  due  to  the  fact  that  the  glass  contracts  slowly  for  a  consider- 
able time  after  being  heated  and  allowed  to  cool,  and  is  thus 
unable  to  follow  immediately  sudden  changes  in  temperature. 
The  depressed  zero  point  is,  however,  the  one  to  be  used  for 
calibration. 

Under  the  supposition  that  the  bore  of  the  tube  and  the  scale 
are  uniform  between  the  two  fixed  points,  the  value  of  a  scale 
part  may  be  found  in  terms  of  the  mercury  in  glass  scale  by 
dividing  t  degrees  by  (t  +  b  —  a).  Calculate  by  this  method  the 
value  of  the  tenth,  twentieth,  thirtieth,  etc.,  divisions  of  the  scale. 
Compare  the  values  so  found  with  a  calibration  obtained  by  direct 
comparison  of  the  thermometer  with  a  standard  thermometer.1 

For  this  comparison  both  thermometers  should  be  placed  in  a 
large  bulk  of  water  in  such  a  way  as  to  keep  their  bulbs  close  to- 
gether, and  the  water  constantly  stirred  during  the  comparison. 
Change  the  temperature  of  the  water  by  steps  of  5°  at  a  time 
from  o°  up  to  50°,  and  read  both  thermometers  each  time  after 
the  water  has  been  well  stirred.  Compare  the  corrected  table 
obtained  by  the  comparison  with  the  standard,  with  the  calibra- 
tion obtained  by  calculation. 

For  accurate  determinations  of  temperature  it  is  necessary  to 
determine  the  depressed  zero  point  immediately  after  taking 
the  temperature  in  question.  The  temperature  must  be  reck- 


*For  the  reduction  of  the  mercury  in  glass  scale  to  the  normal 
(hydrogen)  scale  see  Chappuis,  Rapp.  Congr.  internat.  Paris,  1900,  Vol.  I, 
p.  142. 


100  PHYSICAL    MEASUREMENTS 

oned  from  this  depressed  zero,  even  though  it  is  not  a  constant 
for  different  temperatures. 

Determine  the  fixed  points  of  an  ordinary  thermometer  and 
calibrate  it  from  o°  to  50°  C. 

FORM    OP   RECORD. 

Exercise  37.     To  determine  the  fixed  points  of  a  thermometer. 
Thermometer  No. 


Boil 
Reading 

ing 

point 
Correction 

Zero 
Reading 

point 
Correction 

Calibi 
Compared 

-ation 
Computed 



Value  of  one  scale  part  = 

81.  Stem  Correction.    In  order  that  the  reading  of  the  ther- 
mometer may  give  the  exact  temperature  of  the  body  in  question, 
it  is  necessary  that  the  entire  mass -of  mercury  should  be  at  the 
temperature  of  the  body.     Frequently  a  portion  of  the  filament 
extends  above  the  region  whose  temperature  is  sought,  and  the 
reading  will  be  too  high  or  too  low  according  as  the  temperature 
of  the  filament  is  above  or  below  that  of  the  body  to  be  measured. 
The  relative  coefficient  of  expansion  for  mercury  in  Jena  nor- 
mal glass  is  0.000157  per  degree  C.    If  t°  of  the  filament  are  out- 
side, then  the  difference  in  height  of  the  filament  for  a  difference 
of  i°  C  will  be  0.000157  t°.  If  f°0  be  the  mean  temperature  of  the 
filament  exposed,  and  ^  the  temperature  indicated  by  the  ther- 
mometer, then  the  correction  to  be  added  to  the  indicated  reading 
is  0.000157  (fx — *0)  t  degrees;  t0  is  determined  by  an  auxiliary 
thermometer. 

EXPANSION. 

82.  Coefficient   of   Linear    Expansion.      The    coefficient    of 
linear  expansion  of  a  substance  is  defined  as  the  increase  per  unit 
length,  per  degree  increase  of  temperature.    Thus  if  the  substance 
be  in  the  form  of  a  rod  or  wire,  and  /±  be  its  length  at  temperature 
f±,  and  /2  its  length  corresponding  to  t2,  then  j3,  the  coefficient  of 
linear  expansion,  is  by  definition 


If  one  of  the  temperatures  be  taken  as  o°  C,  the  corresponding 


EXPANSION 


101 


length  /0,  and  the  length  at  any  other  temperature  /,  be  /t,  then  the 
above  formula  reduces  to 


(105) 


or 


(io6) 

83.  Exercise  38.  Coefficient  of  Linear  Expansion  of  a 
Solid.  The  solid  whose  coefficient  of  expansion  is  to  be  deter- 
mined is  in  the  form  of  a  long  thin  rod  R,  Fig.  40,  which  is  held 


Fig.  40. 

by  means  of  thin  rubber  stoppers  S  $,  at  the  center  of  a  brass  tube 
T,  about  five  centimeters  in  diameter,  and  of  approximately  the 
same  length  as  the  rod.  The  tube  is  furnished  with  inlet  and  out- 
let tubes  for  the  passage  of  water  or  steam.  Thermometers  are 
placed  at  the  ends  of  the  tube  in  two  openings  a  a.  The  tube  rests 
on  two  V-shaped  blocks,  which  together  with  the  block  b,  are 
fastened  to  the  stout  board  B.  The  block  b  carries  an  adjustable 
screw  whose  axis  is  in  line  with  that  of  the  rod,  and  which  has  on 
its  inner  end  a  small  glass  cap,  to  prevent  as  far  as  possible  any 
loss  of  heat  from  the  rod  by  conduction.  The  screw  is  turned  till 
the  glass  cap  presses  firmly  against  the  end  of  the  rod,  and  thus 
acts  as  the  fixed  point  from  which  the  length  of  the  rod  is 
measured. 

The  variations  in  length  are  measured  by  means  of  a  series  of 


IO2  PHYSICAL    MEASUREMENTS 

levers  pressing  against  the  other  end  of  the  rod.  The  lever  L, 
mounted  in  a  slot  in  the  block  G,  rotates  about  O,  and  actuates  the 
tilting  mirror  M,  through  the  right  angled  arm  P.  Four  shallow 
grooves,  o' ' ,  o" ,  o'" ,  olv  ,  are  cut  in  the  surface  of  the  block  at  right 
angles  to  the  level  L,  in  which  the  points  of  the  tilting  mirror  M, 
may  be  placed  so  as  to  stand  astride  of  the  slot.  The  rotation  of 
the  mirror  is  read  off  by  means  of  a  telescope  and  a  vertical  scale. 

Loss  of  heat  by  conduction  from  the  rod  is  prevented  by  insert- 
ing a  small  piece  of  glass  between  it  and  the  point  of  the  lever  L, 
while  the  tube  is  covered  with  a  layer  of  asbestos,  and  screens  of 
asbestos  are  placed  at  either  end  to  prevent  loss  by  radiation.  By 
means  of  the  screw  Sc,  the  block  with  its  system  of  levers  is 
pushed  forward  and  the  apparatus  adjusted  so  that,  with  ice  water 
flowing  through  the  tube,  the  mirror  stands  vertically,  and  the 
zero  reading  is  taken. 

If  the  distance  between  0  and  the  point  of  contact  of  the  lever 
with  the  glass  plate  be  called  llt  and  12  be  the  distance  between 
O  and  F,  then  when  the  rod  expands  by  an  amount  a,  for  a  differ- 
ence in  temperature  t,  the  lever  will  rotate  through  an  angle  <£, 
and  its  ends  will  describe  short  arcs  a  and  b,  such  that 

a  =  0/1  (107) 

and 

b  =  $k  (108) 

Also,  since  the  point  F  moves  upward  through  the  short  arc  b, 
the  mirror  is  tilted  through  an  angle  0,  such  that 

b  =  op  (109) 

where  p  is  the  distance  o'  F. 

By  eliminating  b  and  </>  from  the  above  equations  we  have 

9=  ^  •  (no) 

/>/! 

Since  the  quantity  l2/p  /x  is  a  constant  of  the  instrument  we  may 
write 

0  =  ka  .  (in) 

The  angle  0  is  evaluated  from  the  reading  a,  observed  in  the 


EXPANSION  IO3 

telescope,  and  the  distance  D,  between  the  telescope  and  the  mir- 
ror by  means  of  equation  (18). 


Since  k,  a,  and  D  are  known,  the  value  of  a  may  be  readily 
determined,  and  from  the  relation 


(112) 
we  have 


Determination  of  k.  To  determine  the  constant  k,  put  glass 
slides  of  known  thickness  pf  measured  by  means  of  the  sphero- 
meter,  between  the  end  of  the  rod  and  the  lever  and  observe  the 
resulting  deflection,  while  the  temperature  and  the  position  of  the 
block  G,  remain  unchanged.  Solve  for  k  from  the  relation 


(114) 


Use  two  different  slides  and  two  positions  of  the  mirror.  Since 
for  accurate  determinations  of  k,  it  is  desirable  that  the  glass  slides 
should  not  be  too  thin,  the  telescope  and  scale  should  be  brought 
nearer  to  the  mirror,  in  order  to  keep  the  readings  on  the  scale. 
In  determining  the  deflection  during  the  experiment,  the  sensi- 
tiveness may  be  increased  at  will  by  increasing  the  distance  be- 
tween the  mirror  and  scale. 

The  temperature  of  the  rod  is  changed  by  passing  water  of  the 
desired  temperature  from  a  larger  vessel  through  the  tube.  As 
soon  as  the  temperature  becomes  constant,  as  indicated  by  the  con- 
stancy of  the  scale  reading,  take  this  reading  and  that  of  the  ther- 
mometers. Thermometer  readings  must  be  corrected  for  the  ex- 
posed filament.  Change  the  temperature  of  the  rod  by  steps  of 
about  20°  from  o°  to  100°  C.  In  order  to  obtain  the  last  point 
steam  is  passed  through  the  tube  and  the  barometer  reading 
noted.  Since  the  change  in  length  is  a  very  small  fraction  of  the 


IO4 


PHYSICAL    MEASUREMENTS 


original  length  of  the  rod,  this  length  may  be  determined  with 
sufficient  accuracy  by  means  of  a  millimeter  scale. 

Since  in  the  instrument  as  described  above  a  small  portion  of 
the  rod  at  each  end  is  not  at  the  exact  temperature  of  the  water 
or  steam,  a  slight  error  will  be  introduced,  especially  if  the  differ- 
-ence  between  the  temperature  of  the  rod  and  that  of  the  room  be 
large.  The  determination  may  be  made  much  more  accurate  by 
holding  the  rods,  which  are  cut  slightly  shorter  than  the  tube,  in 
the  center  of  the  tube  by  wire  frames,  and  inserting  in  the  rubber 
stoppers  short  pieces  of  nickel-iron  whose  coefficient  of  expansion 
is  almost  negligible. 

In  a  similar  type  of  apparatus  the  block  with  the  system  of 
levers  is  replaced  by  a  spherometer  screw,  passing  through  a 
nut  on  a  support  solidly  fastened  to  the  base  of  the  instrument. 
The  end  of  the  screw  is  brought  into  contact  with  the  movable 
end  of  the  rod  and  the  position  of  the  screw  is  read  from  the 
spherometer  scale.  Readings  are  repeated  with  the  rod  at  differ- 
ent temperatures.  From  these  readings  the  expansion  of  the  rod 
can  easily  be  calculated.  The  form  of  record,  given  below,  must 
of  course  be  altered  correspondingly,  if  this  type  of  apparatus  be 
used. 

FORM   OF   RECORD. 

Exercise  38.  To  determine  the  coefficient  of  linear  expansion 
•of  a  metal  and  of  glass. 

Date 

I.     Determination  of  k. 

Thickness  of  glass  slide a          D  tan  2  &  0          k 


II.     Determination  of  /?.     Length  of  rod, 

Temperature  a  D  tan  2  0 


84.  Expansion  of  Liquids.  In  determining  the  coefficient 
of  expansion  of  a  liquid,  we  are  met  by  the  difficulty  that  the 
liquid  must  be  contained  in  some  sort  of  a  receptacle,  the  material 
of  which  expands  at  the  same  time  as  the  liquid,  and  hence  the 
apparent  expansion  of  the  liquid  is  always  a  differential  effect, 


EXPANSION  IO5, 

being  the  difference  between  the  increase  in  volume  of  the  recep- 
tacle and  that  of  the  expanding  liquid.  By  measuring  the  height 
of  two  communicating  columns  of  the  same  liquid  at  different 
temperatures  the  coefficient  of  cubical  expansion  of  the  liquid  may 
be  determined  without  reference  to  the  expansion  of  the  contain- 
ing vessel.  For  the  attainment  of  accurate  results,  however,  the 
apparatus  becomes  too  complicated  for  use  in  an  elementary 
course1. 

It  is  therefore  customary  to  measure  the  relative  expansion 
of  the  liquid  in  a  vessel,  usually  of  glass,  and  calculate  a  the 
absolute  coefficient  of  expansion  of  the  liquid  from  this  value 
and  from  the  known  coefficient  of  cubical  expansion  g  of  the  vessel. 
This  latter  coefficient  is  found  either  by  determining  {$,  the  linear 
coefficient  for  the  glass  of  which  the  vessel  is  made,  and  putting 
a  —  3  /?,  or  by  using  a  liquid  whose  absolute  coefficient  is  known. 
Thus  let 

a  be  the  relative  coefficient  of  cubical  expansion  of  the  liquid  in 

glass, 

a  its  absolute  coefficient, 

g  the  coefficient  of  cubical  expansion  for  glass, 
Fthe  apparent  volume  of  the  liquid  at  t°  C, 
V\  the  real  volume  at  t°  'C, 
Fo  its  volume  at  o°  C. 


Then  we  have  the  following  relations  : 


and 


from  which  by  eliminating  F±  and  t,  we  have 

a  =  a  +  g^,l  ("8) 

KO  .-    , 

or,  since  F/F0  is  very  nearly  unity 


1  Preston,  Theory  of  Heat,  p.  170. 


106  PHYSICAL    MEASUREMENTS 

85.  Exercise  39.  Coefficient  of  Expansion  of  a  Liquid  by 
the  Dilatometer.  A  dilatometer  consists  of  a  bulb  with  an  ac- 
curately graduated  stem  of  uniform  capillary  bore. 
In  work  of  extreme  precision  it  is  necessary  to 
calibrate  the  stem  throughout  in  order  to  correct  for 
irregularities  of  cross  section.  The  volume  of  the 
bulb  at  o°  C,  and  that  of  one  scale  division  on  the 
stem  must  be  accurately  determined,  as  well  as  the 
coefficient  of  cubical  expansion  of  the  bulb.  These 
form  the  constants  of  the  instrument.  The  exercise 
is  divided  into  three  parts  : 

(a;)  Determination  of  the  volume  of  one  division 
of  the  capillary  stem.  Bring  into  the  graduated  stem 
an  amount  of  mercury  sufficient  to  fill  it  nearly  full, 
and  measure  the  length  of  the  filament  in  terms  of 
the  scale  divisions,  using  a  mirror  scale  or  reading 
microscopes.  Let  the  observed  length  of  the  filament 
be  /  scale  divisions.  On  account  of  the  meniscus  the 
length  measured  from  end  to  end  will  be  slightly  too 
large.  For  capillary  tubes  it  is  sufficient  to  subtract 
from  the  observed  reading  0.4  h,  where  h  is  the 


height  of  the  meniscus  in  terms  of  a  scale  division. 
Let  the  mass  of  the  mercury  filament  be  m  grams,  at  the 
temperature  t°.  One  gram  of  mercury  weighed  in  air,  occupies  at 
t°  C,  the  volume 

v'  —  0.07355  (i  -j-  0.000181  0  cm3  (120) 

so  that  the  volume  corresponding  to  one  scale  division  is 


(b)  Determination  of  the  volume  of  the  bulb.  The  volume 
of  the  bulb  may  be  determined  by  taking  the  mass  of  the  dilatom- 
eter, first  when  empty  and  dry,  and  secondly  when  filled  with 
air-free  water  up  to  scale  part  /'  in  the  capillary  tube,  at  a 


EXPANSION 


temperature  not  below  15°  C.     Let  the  mass  in  the  first  case  be 
iii0,  and  in  the  second  wt,  then  m,  the  mass  of  the  water,  is 


m  =  mr  —  11 t< 


One  gram  o'f  water,  weighed  with  brass  weights  in  air,  occupies 
very  nearly  (2.00106 — •  d)  cm3,  where  d  is  the  density  of  the 
water  at  temperature  t° ,  as  given  in  Table  II.  The  volume  of  the 
bulb  and  the  capillary  tube  up  to  the  zero  mark  of  the  stem,  at  this 
temperature  is  therefore, 

J/i  =  [w(2.ooio6  —  d)~  V  v]  cm3  (121) 

The  volume  F0,  at  o°  is  readily  found  from  this  by  applying 
the  equation 


where  g  may  be  taken  as  0.000025. 

If  it  be  desired  to  determine  the  coefficient  of  cubical  expansion 
of  the  glass  vessel  more  accurately,  it  is  best  to  fill  the  dilatometer 
with  pure  mercury  and  determine  the  readings  l\  and  l\  at  two 
different  temperatures,  say  t°  and  o°  C.  If  M  be  the  mass  of  the 
mercury,  and  the  values  of  i/  (equation  t2o)  at  the  two  tempera- 
tures v\  and  z"'0,  then 

V,  =  Mv\  —  l\v  and  V*  =  Mi/*  —  l'*v 

from  which  g  can  easily  be  calculated. 

(c)  Coefficient  of  expansion  of  a  liquid.  Fill  the  dilatometer 
with  the  liquid  under  investigation  and  immerse  it  in  a  large  glasi 
vessel  containing  water.  Vary  the  temperature  of  this  bath  and 
observe  the  resulting  height  of  the  liquid  in  the  stem  of  the  instru- 
ment. Be  sure  to  leave  the  instrument  in  the  bath  long  enough  to 
insure  thermal  equilibrium  between  its  contents  and  the  bath. 
Stir  constantly  to  keep  the  temperature  of  the  bath  uniform  in  all 
parts.  Add  small  quantities  of  warm  or  cold  water,  if  necessary, 
to  keep  the  temperature  of  the  bath  constant. 


loS 


PHYSICAL    MEASUREMENTS 


Take  four  or  five  different  temperatures,  tlf  tzt  ts, ,  and 

observe  the  corresponding  readings  on  the  stem,  lit  I2f    

Denote  the  related  volumes  by  V^,  VZ) ,then 


etc. 


(122) 


Plot  volumes  and  temperatures,   and  calculate  the  mean  co- 
efficient of  expansion  from 


(123) 


In  case  the  expansion  of  water  is  to  be  studied  in  the  neighbor- 
hood of  4°  C,  the  bulb  must  be  quite  large  or  the  capillary  tube 
of  smaller  bore  than  in  the  average  instrument,  as  the  expansion 
of  water  at  these  temperatures  is  very  slight.  Plot  the  apparent 
and  the  real  volumes  as  ordinates  and  the  temperatures  as  abscis- 
sae. Compute  a  table  for  the  specific  volumes,  V/M,  for  the  dif- 
ferent temperatures. 

FORM    OF   RECORD. 

Exercise  39.  To  determine  the  coefficient  of  expansion  of  a 
liquid  by  means  of  the  dilatometer. 


Date. 


(a)     Value  of  one  scale  division  of  stem. 

•Mass  of  mercury  —....,  number  of  divisions  =  . . . .,  v 

(fc)     Volume  of  bulb  at  o°  C. 

Mass  of  dilatometer  empty ,  Temperature. . 

'Mass  of  dilatometer  filled ,  m— , 

I'  = 


'Coefficient  of  expansion  of  the 

•Mass  of  dilatometer  filled 

Mass  oi  dilatometer 


lass. 
Reading 


V 


(c)     Coefficient  of  expansion  of 
Reading  t 


V 


EXPANSION 


109 


86.  Air-Free  Water.     Air-free  water  is  used  in  many  cases 
for  finding  the  volume  of  glass  instruments.     Such  water  may  be 

prepared  by  boiling  distilled  water  for 
half  an  hour.  Another  method  for  free- 
ing water  from  the  absorbed  air  is  shown 
in  Fig.  42.  The  water  is  contained  in  a 
flask  which  is  closed  by  a  rubber  stopper 
through  which  passes  a  capillary  tube  of 
fine  bore,  extending  almost  to  the  bottom 
of  the  vessel.  A  water  aspirator  con- 
nected to  the  side  tube  produces  a  partial 
vacuum  in  the  flask  and  air  is  forced 
in  through  the  capillary  tube  and  rises 
through  the  water.  This  air,  rising 
through  the  water  under  diminished  pres- 
sure, produces  a  state  of  unstable  equi- 
librium in  the  air  previously  absorbed 
in  the  water  and  this  absorbed  air  can 
now  be  seen  rising  in  the  form  of  small 
bubbles.  The  process  can  be  hastened  by 
giving  the  flask  from  time  to  time  a 
smart  blow  with  the  hand. 

87.  Exercise  40.     Constant  Volume  Air  Thermometer.     If 
the  volume  of  a  given  mass  of  gas  be  kept  constant  and  its 
temperature  be  varied,  the  pressure  of  the  gas  upon  the  walls  of 
its  containing  vessel  will  vary  according  to  the  formula 

p t  =  p0(i -{- a  t) .  (124) 

where  p0  and  pt  are  the  pressures  at  o°  and  t°  C,  respectively, 
and  a  is  the  pressure  coefficient  of  the  gas  at  constant  volume. 
We  know  that  for  a  perfect  gas  a  is  the  same  as  the  coefficient  of 
voluminal  expansion  for  the  gas  at  constant  pressure.1  For  all 
permanent  gases  the  value  of  the  pressure  coefficient  is  nearly 
the  same.  In  order  to  determine  a,  the  gas,  usually  air,  is  kept  at 
constant  volume,  its  temperature  is  varied  and  the  resulting  pres- 


Fig.  42. 


1  College  Physics,  Article  163. 


no 


PHYSICAL,    MEASUREMENTS 


sures  observed.  When  properly  calibrated  the  instrument  furn- 
ishes the  means  of  determining  temperatures  as  a  function  of  the 
observed  pressures,  and  hence  is  termed  a  constant  volume  air 
thermometer. 

The  air  is  contained  in  the  bulb  A,  Fig.  43,  which  is  attached 
to  a  fine  capillary  tube  a,  bent  twice  at  right  angles,  and  joined  at 
its  outer  end  to  a  larger  tube  B  about  one  centimeter  in  diameter. 


Fig.  43- 


The  tube  B  has  at  the  point  of  junction  with  the  capillary  tube  a 
fine  pointer  p  of  colored  glass  fused  into  its  side,  which  serves  as  a 
point  of  reference  for  the  height  of  the  mercury  column  in  B,  and 
to  which  the  mercury  must  be  brought  each  time  before  a  reading 
is  taken.  A  mark  on  the  capillary  may  serve  the  same  purpose. 
In  the  simpler  forms  of  the  apparatus  the  -tube  B  is  connected 
by  a  thick  walled  rubber  tube  to  a  second  glass  tube  C,  which  is 
capable  of  considerable  movement  up  and  down  a  graduated  scale 
placed  between  the  two  tubes.  By  means  of  this  scale  the  level  of 


EXPANSION  1  1  1 

the  mercury  in  each  tube  may  be  read  off  and  the  corresponding 
pressure  at  any  temperature  upon  the  gas  determined.  In  this 
way  the  volume  of  the  enclosed  air  is  kept  constant  except  for  the 
change  in  volume  of  the  bulb  due  to  expansion,  which  must  be 
allowed  for  in  the  final  computation. 

In  order  to  determine  the  pressure  coefficient  of  dry  air  at 
constant  volume,  the  bulb  is  first  carefully  cleaned,  dried,  filled 
with  dry  air,  and  surrounded  with  melting  ice.  The  tube  C  is 
then  moved  up  or  down  until  the  surface  of  the  mercury  in  B  just 
touches  the  tip  of  the  colored  pointer.  In  this  position  the  tube  C 
is  clamped  and  the  reading  on  the  surface  of  the  mercury  in  each 
tube  is  noted.  If  the  difference  in  these  readings  be  h0,  and  we 
call  the  barometric  reading  b,  then  the  pressure  upon  the  gas  will 
correspond  to  a  barometric  height 


(115) 


The  bulb  is  next  surrounded  by  steam  and  the  tube  C  adjusted 
so  as  to  bring  the  mercury  again  to  the  tip  of  the  pointer,  and  a 
second  reading  taken.  Care  must  be  taken  to  wait  long  enough 
before  reading  to  allow  the  air  in  the  bulb  to  reach  the  temperature 
of  the  steam,  a  condition  clearly  indicated  by  the  level  of  the  sur- 
faces of  the  mercury  columns  remaining  constant.  If  we  denote 
the  difference  in  level  between  the  two  columns  by  h,  the  air  is 
under  a  pressure  corresponding  to  a  barometric  height 

H  =  b  +  h.  (126) 

In  measuring  b  and  h  it  is  sufficient  to  reduce  both  to  the  same 
temperature  throughout  the  experiment,  but  for  finding  t,  the 
temperature  of  the  steam,  it  is  necessary  to  reduce  the  barometric 
reading  to  o°  C,  in  order  to  make  use  of  Table  IX.  In  all  ma- 
nipulations of  the  instrument  great  care  must  be  exercised  to  pre- 
vent any  mercury  from  entering  the  capillary  tube,  and  being 
drawn  into  the  bulb. 


112 


PHYSICAL    MEASUREMENTS 


From  Gay  Lussac's  law1  deduce  the  following  formula  for  the 
pressure  coefficient  of  a  gas  at  constant  volume  where  g  denotes, 
as  usual,  the  coefficient  of  voluminal  expansion  for  the  glass 
bulb:2 


—  H. 


H 


What  is  the  formula  for  any  temperature  t,  as  given  by  the  air 
thermometer  ? 

Insert  the  thermometer  in  water  of  different  temperatures  and 
compare  the  calculated  temperatures  with  those  measured  by  a 
mercury-in-glass  thermometer.  Plot  temperatures  and  pressures. 

In  the  derivation  of  the  above  formula  it  is  assumed  that  the 
volume  of  the  capillary  connection  from  the  bulb  to  the  tip  of  the 
pointer  is  negligible  as  compared  to  the  volume  of  the  bulb.  This 
assumption  is  the  more  nearly  justified  the  larger  the  bulb  and  the 
finer  the  bore  of  the  capillary  tube. 


FORM    OF   RECORD. 


Exercise  40.     To  determine  the  pressure  coefficient  of  dry  air 
by  means  of  the  constant  volume  air  thermometer. 


Barometric  pressure. 


Temperature 


Readinj 
B 


Pressure 
in  cm  of  Hg 


CALORIMETRY. 

88.  Definitions.  Temperature  is  to  be  sharply  distinguished 
from  quantity  of  heat.  The  former  has  reference  only  to  the 
condition  of  a  body  affecting  the  sensation  of  warmth  and  cold 
and  has  no  reference  to  the  amount  of  matter  involved.  In 
quantity  of  heat  account  must  be  taken  both  of  the  temperature 


1  College  Physics,  Article  161. 

2  For  the   derivation  of   complete    formula,   see   Kohlrausch,   Physical 
Measurements,  3d  English  from  7th  German  ed.,  pp.  93-97. 


CALORIMETRY  113 

of  the  body  and  of  its  mass.  The  unit  of  temperature  is  the  de- 
gree centigrade.  The  unit  of  quantity  of  heat  is  the  calorie.  A 
calorie  is  the  quantity  of  heat  required  to  raise  the  temperature  of 
one  gram  of  water  i°C.  The  thermal  capacity  of  a  body  is 
numerically  equal  to  the  number  of  calories  required  to  raise  the 
temperature  of  the  body  one  degree  centigrade.  The  thermal 
capacity  of  the  substance,  of  which  the  body  is  composed,  is  its 
thermal  capacity  per  unit  mass,  or  it  is  numerically  equal  to  the 
heat  in  calories  required  to  raise  the  temperature  of  one  gram  of 
the  substance  one  degree  centigrade.  If  we  call  the  thermal 
capacity  of  a  substance  c,  the  mass  of  the  bod)  M,  then  the  heat 
needed  to  raise  the  temperature  of  the  body  from  t.2  to  t±  degrees 
is  given  by 

H  =  cM(h  —  fc)  calories  (  128) 


From  the  definition  of  the  calorie  it  follows  that  the  thermal 
capacity  of  water  is  taken  as  unity.  Though  c  for  water  varies 
slightly  with  temperature1,  it  will  be  considered  as  constant  in  the 
following  exercises. 

The  specific  heat,  s,  of  a  substance  is  the  ratio  of  the  thermal 
capacity  of  the  substance  to  that  of  water,  or 


(129) 


Since  cw  is  unity  the  specific  heat  of  a  substance  is  numerically 
equal  to  its  thermal  capacity. 

The  specific  heat  of  any  substance  is  different  for  different 
temperatures  and  hence  as  usually  given,  it  denotes  the  mean  value 
for  the  specific  heat  between  certain  temperature  limits. 

89.  Specific  Heat  by  Method  of  Mixtures.  If  two  sub- 
stances of  masses  M^  and  M2,  at  temperatures  £±  and  t2  and  having 
thermal  capacities  c±  and  c.2f  be  brought  into  contact,  they  will 
come  to  some  intermediate  temperature  t,  such  that  the  number 
of  calories  given  out  by  the  first  is  exactly  equal  to  the  number 


^College  Physics,  Article  i/i. 


114 


PHYSICAL    MEASUREMENTS 


gained  by  the  second,  provided  of  course,  that  no  heat  has  been 
added  from  the  outside,  or  been  lost  externally  through 
conduction  or  radiation.  Then  the  equation  for  the  heat  ex- 
change is 

t  —  tz).  (130) 


If  the  second  substance  be  water,  then  M2  — 
the  expression  becomes 


= cw  and 
(131) 


In  actual  practice  it  is  impossible  to  avoid  loss  of  heat  both  by 
radiation  and  by  conduction.  If  the  water  be  contained  in  a  vessel 
or  calorimeter,  then  the  latter  receives  heat  along  with  the  water 
and  finally  comes  to  the  common  temperature  /.  If  M  be  the 
mass  of  the  calorimeter  and  c  its  thermal  capacity  the  amount  of 
heat  needed  to  raise  its  temperature  from  f,  to  t  degrees  is 

H  =  cM(t  —  f8) 

This  quantity  must  be  added  to  the  right  hand  member  of  equa- 
tion (130).  The  calorimeter  usually  consists  of  a  beaker,  stirrer 
and  thermometer  for  each  of  which  cM  must  be  calculated  and 
their  sum  be  taken  in  the  above  equation.  If  we  set 


=  Z  —  M  =  2  sM 


(132) 


then  m  is  the  mass  of  water  which  has  the  same  thermal  effect  as 
the  calorimeter.  This  mass  is  called  the  water-equivalent  of  the 
calorimeter.  Therefore  the  total  heat  gained  by  the  water  and 
calorimeter  will  be  cw  (m  +  Mw)  (f  —  *2)  and  we  have 


M,)  (f  —  fQ 


d33) 


90.     Exercise    41.      Water-Equivalent    of    a     Calorimeter. 
The  calorimeter  is  a  cup  of  thin  metal,  preferably  of  aluminium. 


CAI.ORIMETRY  115 

which  is  placed  inside  a  large  vessel  upon  a  flat  piece  of  cork  or 
other  poor  conductor.  In  the  calorimeter  are  a  stirrer  and  a  ther- 
mometer. Let  m  be  the  water-equivalent  in  grams  of  the  calorim- 
eter including  stirrer  and  thermometer;  also  let  the  calorimeter 
contain  Mw  grams  of  water  at  a  temperature  t2 ;  suppose  the  re- 
sulting temperature,  due  to  adding  M±  grams  of  water  at  tlf  finally 
comes  to  be  t.  Then  the  exchange  of  heat  is  represented  by  the 
equation 

<-'w(^w  +  "0  (fa  — 0  =rwMi  (/  — fi)  (134) 


or,  solving  for  m 


—  M, 


(135) 


In  practice  weigh  the  calorimeter  empty  and  dry,  then  fill  about 
one-third  full  with  water  at  a  temperature  about  fifteen  degrees 
above  the  temperature  of  the  room  and  weigh  again.  The  differ- 
ence is  Mw.  Next  add  water  at  a  temperature,  about  ten  degrees 
below  room  temperature,  until  the  resulting  temperature  after  vig- 
orous stirring  is  about  room  temperature.  The  temperature  of 
the  cold  and  warm  water  should  be  carefully  determined,  just 
before  mixing,  by  means  of  a  thermometer  reading  to  tenths  of  a 
degree  centigrade.  The  resulting  temperature  is  to  be  taken  only 
after  the  thermometer  reading  has  become  constant.  Repeat  the 
experiment  twice,  taking  the  mean  of  the  three  results  as  the 
water-equivalent.  For  the  correction  due  to  radiation  see  Art.  92. 


FORM    OF   RECORD. 


Exercise  41.     To  determine  the  water-equivalent  of  a  calori- 
meter. 


Weight  of    Vessel  with 
vessel         l   water   1st 

Mw 

t* 

fi 

t     Vessel  with 
water    2d 

MX 

m 

Compare  this  result  with  that  obtained  by  multiplying  the  mass 
of  the  calorimeter  by  the  specific  heat  of  the  metal,  as  given  in 
Table  X.  In  the  case  of  a  thermometer  the  exact  masses  of  mer- 


IIO  PHYSICAL    MEASUREMENTS 

cury  and  glass  are  unknown.  But  the  thermal  capacities  for 
equal  volumes  of  mercury  and  glass  are  nearly  the  same,  namely 
0.47  calories  per  cc.  To  find  the  water-equivalent  of  a  thermom- 
eter it  is  therefore  sufficient  to  multiply  the  immersed  volume  of 
the  thermometer  by  this  number. 

91.  Exercise  42.  Specific  Heat  of  Copper.  The  piece  of 
copper  whose  specific  heat  is  to  be  determined  is  heated  in  a  brass 
tube  (Fig.  44),  which  is  surrounded  by  a  steam  jacket.  The 
copper  is  hung  by  a  thread  in  the  middle  of  the  tube  and  the 
top  is  closed  by  means  of  a  cork  carrying  a  thermometer.  The 
heater  sits  upon  a  wooden  board  having  a  hole  of  the  diameter 
of  the  inner  tube  and  directly  beneath  it.  This  board  slides  upon 


Fig.  44. 

a  support  provided  with  a  similar  hole  and  so  arranged  that  the 
two  holes  coincide  when  the  board  is  pushed  in  as  far  as  possible. 
Immediately  under  the  hole  in  the  support  is  placed  the  calori- 
meter so  that  the  heated  body  may  be  passed  directly  from  the 
tube  through  the  support  into  the  calorimeter.  The  sliding  board 
is  to  shield  the  calorimeter  from  heat  during  the  heating  of  the 
copper. 

In  use  the  dry  copper  is  weighed  and  hung  in  place,  the  hole  in 


CALORIMETRY 


117 


the  support  closed  by  the  sliding  board  and  steam  passed  through 
the  jacket  until  the  temperature  of  the  interior  becomes  constant, 
t^.  This  heating  usually  requires  about  twenty  minutes.  Mean^ 
while  the  calorimeter  with  the  contained  water  is  carefully 
weighed.  The  temperature  of  the  water  in  the  calorimeter  is 
then  read,  t2.  The  calorimeter  is  put  in  position,  the  sliding 
board  pushed  in,  and  the  heated  copper  lowered  gently  into  the 
vessel  beneath.  The  calorimeter  with  its  contents  is  then  re- 
moved, the  water  thoroughly  stirred  and  the  highest  temperature  t, 
carefully  noted.  In  order  to  avoid  the  necessity  of  correcting  for 
radiation,  it  is  well  to  have  the  temperature  of  the  water  in  the 
calorimeter  some  4°  or  5°  below  the  temperature  of  the  room  at 
the  beginning  the  experiment.  Apply  formula  (133). 

FORM    OF   RECORD. 

Exercise  42.     To  determine  the  specific  heat  of  copper. 


Date. 

Mass  of 

Calorimeter 

Mi 

M* 

m 

t* 

t 

Calorimeter 

with  water 

Specific  heat  = 


92.  Correction  for  Radiation.  In  the  preceding  experiments 
the  temperatures  were  so  chosen  as  to  render  correction  for 
radiation  and  absorption  unnecessary.  In  experiments  requiring 
a  greater  degree  of  accuracy  this  loss  or  gain  of  heat  by  the  cal- 
orimeter must  be  taken  into  account.  This  is  best  done  by  noting 
times  and  temperatures  for  an  interval  of  at  least  five  minutes  be- 
fore the  instant  at  which  the  experiment  proper  begins,  that  is,  the 
instant  at  which  the  ice,  steam  or  metal  enters  the  calorimeter. 
Readings  should  be  taken  every  twenty  seconds.  Plot  the  times  as 
abscissae  and  the  temperatures  as  prdinates,  (Fig.  45).  After  the 
beginning  of  the  experiment  the  temperature  rapidly  rises  or  falls, 
and  having  reached  a  maximum  or  a  minimum  it  will  practically 
become  a  linear  function  of  the  time  for  a  few  minutes,  showing 
the  rate  of  radiation  or  absorption. 


iiS 


PHYSICAL  MEASUREMENTS 


According  to  Newton's  law  of  cooling,  the  rate  of  loss  or  gain 
of  heat  energy  by  a  body  due  to  radiation,  varies  directly  as  the 
difference  in  temperature  between  the  body  and  surrounding  ob- 
jects. From  the  straight  parts  of  the  curve  determine  this  rate  r, 
for  a  difference  in  temperature  of  one  degree.  Next  determine 
rlie  average  temperature  of  the  calorimeter  for  the  time  interval 
(abt  Fig.  45),  during  which  the  mixing  occurs.  This  is  done  by 
dividing  ab  into  a  number  of  small  intervals,  taking  the  sum  of  the 


Minutes 

Fig.  45- 


temperatures  belonging  to  each,  and  dividing  by  the  number  of  in- 
tervals. The  difference  between  this  average  and  room  tempera- 
ture multiplied  by  the  rate  r,  gives  the  correction  to  be  applied 
with  its  proper  sign,  to  the  observed  reading  at  the  time  b. 

A  numerical  example  will  make  the  method  clearer. 

The  readings  of  the  thermometer  were  begun  ten  minutes  before 


CALORIMETRY 


119 


the  heated  body  was  introduced  into  the  calorimeter,  and  the  read- 
ing continued  for  twenty  minutes  after  mixing  took  place.  The 
room  temperature  was  19°. 45  C. 

Read. 

2i°.97 
.94 
.92 
.89 
.87 
-84 
.83 
•79 
•76 
•74 
•71 


Min.         Read.      >Min.         Read            Aver. 

Min. 

o            i8°.04         10          i8°.i9          (Calc.) 
i                 .06         ii           19  .8              19  .0 

20 
21 

2                       .07             12              21    .O 

20  .4 

22 

3                  -09          13           21  .7 

21   1 

23 

4                  .10          14           21  .9 

21    .8 

24 

5                         -12             15               22   .0 

6                 .13         16          22  .04 

21    .95 
22   .02 

25 
26 

7                  -15          17          22  .04 

.16             l8              22   .02 

22   .04 
22   .03 

27 
28 

9                        .17             19               22   .00 
10                                         20              21   .97 

22   .OI 
21    .98 

29 

30 

Average  temperature, 

Before  mixing:  i8°.i2  C. 

2Oth  to  3oth  minute:  21°.  83  C 

• 

Rate  of  change  of  temperature  per  minute  : 

"Rpfnrp   mivino-.                  Io.I7 

18.04 

o.°oi 

20th  to  30th  minute:    ?±£L — 2171  _0o>026C,p,erminute> 

10 


Rate  per  minute  per  degree, 
Before  mixing:  °-OI44 

2oth  to  3oth  minute : 


19.45  —  18.12 
0.026 


=  o°.on  C  per  minute  per  degree. 
=  o°.on  C  per  minute  per  degree. 


21.83  —  1945. 

Average  temperature  of  the  calorimeter  from  the  loth  to  2Oth 
minute : 

2i°.46C. 

Decrease  in  temperature  due  to  radiation  in  10  minutes, 

10(21.46  —  19.45)0.011  =  0.°22  C 

Highest  temperature  corrected  for  radiation : 

21.97  +  0.22  =  22°. 19  C. 
Total  temperature  change,  corrected : 

22.19—  18.19  r=4°.ooC. 


J2O  PHYSICAL    MEASUREMENTS 

93.  Exercise  43.  Heat  of  Fusion  of  Water.  Water  absorbs 
definite  amounts  of  heat  energy  on  passing  from  the  solid  to  the 
liquid,  and  from  the  liquid  to  the  gaseous  state.  The  quantities  of 
heat  per  gram  thus  absorbed  are  termed  the  heat  of  fusion  and 
the  heat  of  vaporization.  The  number  of  calories  necessary. 
to  change  one  gram  of  ice  at  o°C.  to  water  at  o°C.  is  therefore 
numerically  equal  to  the  heat  of  fusion  of  water.  This  may  be 
measured  in  various  ways.  One  of  the  simplest  is  by  the  method 
of  mixtures  ;  i.  e.,  a  known  mass  of  ice  at  o°,  is  added  to  a  definite 
mass  of  water  at  a  known  temperature,  and  the  temperature  of 
the  water  at  the  end  of  the  melting  enables  us  to  compute  the 
amount  of  heat  consumed  in  melting  the  ice. 

Thus  let  M2  grams  of  ice  at  zero,  be  added  to  M±  grams  of 
water  at  ^  and  let  the  temperature  at  the  end  of  the  melting  be 
t;  also  let  the  water-equivalent  of  the  calorimeter,  stirrer  and 
thermometer  be  m,  and  let  /  denote  the  heat  of  fusion  of  ice  as  de- 
fined above.  Then  since  the  water  formed  by  the  melting  of  the  ice 
must  be  warmed  to  t  degrees,  we  have  the  heat  lost  by  the  cal- 
orimeter and  its  contents  equal  to  the  heat  absorbed  by  the  ice  and 
the  ice-water.  Hence 

M2/  +  rwM2*  =  fw(M,  +  m)  (*i— 0  (136) 

from  which  since  cw  equals  unity,  we  have  the  numerical  equality 

i=  w.+g  ".-o_,.  (I37) 

.  The  apparatus  consists  of  a  calorimeter,  a  thermometer,  and  a 
circular  stirrer  covered  with  wire  gauze  to  keep  the  pieces  of  ice 
under  water  while  melting.  Weigh  the  calorimeter,  fill  nearly 
full  of  water  at  a  temperature  tlt  about  fifteen  degrees  above  room 
temperature  and  weigh  again.  The  difference  is  the  mass  of 
water  M±.  Break  clean  ice  into  small  pieces  and  add  to  the  water 
sufficient  dry  ice  to  bring  the  temperature  of  the  calorimeter  and 
its  contents  to  about  fifteen  degrees  below  room  temperature, 


CALORIMETRY 


12  T 


when  all  the  ice  is  melted.  Stir  vigorously  throughout  the  opera- 
tion, read  the  temperature  t,  as  soon  as  the  ice  is  all  melted,  and 
weigh  the  calorimeter  and  its  contents  once  more.  The  differ- 
ence between  the  last  two  weighings  gives  the  mass  of  the  ice  Mz, 
that  was  added. 

FORM    OF  RECORD. 

Exercise  43.     To  determine  the  heat  of  fusion  of  water. 


w 

Alone 

eighings  of 
With   Mt 

calorimeter  : 
With  M!  +  Ma 

Ml 

Dal 
Ma 

e.  . 

m 

ti 

t 

Heat  of  fusion  of  water  = 


94.  Exercise  44.  Heat  of  Vaporization  of  Water  at  Boil- 
ing Point.  The  heat  of  vaporization  at  the  boiling  point  is  the 
number  of  calories  per  gram  required  to  change  water  at  that 
temperature  into  steam  at  the  same  temperature.  Conversely  if 
one  gram  of  steam  at  this  temperature  be  condensed  into  water 
the  same  number  of  calories  will  be  liberated.  Thus  if  M2  grams 
of  steam  at  a  temperature  t2,  having  been  conducted  into  a  cal- 
orimeter of  water  equivalent  m,  containing  M±  grams  of  water  at 
temperature  t^  produce  by  condensation  and  cooling  .a  result- 
ant temperature  t,  we  may  write  our  equation  of  heat  thus : 


M2  L 


or,  since  cw  equals  unity 


-j-  m)  (f  — 


(138) 


(139) 


where  L  is  the  heat  of  vaporization  of  water.  The  water  equiva- 
lent m,  of  the  calorimeter,  may  be  found  experimentally  as  before, 
or  by  multiplying  the  mass  of  the  calorimeter  by  its  specific  heat. 

The  determination  may  be  made  by  either  of  the  following 
methods : 

First  Method.     Steam  generated  in  a  suitable  flask  is  passed 


122 


PHYSICAL    MEASUREMENTS 


Asbestos 
Wood- 
Fig.  46. 


through  a  wide  tube  15  cms  long  and  3  cms  wide,   (Fig.  46), 
in  which  the  water  from  condensation  is  caught  and  retained. 

From  this  it  passes  directly  into 
the  water  in  the  calorimeter.  The 
mass  of  steam  condensed,  M2,  is 
determined  from  the  increase  in 
weight  of  the  calorimeter.  The 
calorimeter  having  been  care- 
fully dried  and  weighed  is  filled 
nearly  full  of  water  at  a  tempera- 
ture some  fifteen  or  twenty  de- 
grees below  that  of  the  room  and 
again  weighed.  The  difference 
in  weight  is  M±.  After  the  steam 
passes  freely  from  the  vertical  tube  leading  from  the  water  trap, 
the  calorimeter  and  its  contents  are  brought  into  place  and  the 
steam  passed  directly  into  the  water  until  its  temperature  is  some 
fifteen  or  twenty  degrees  above  the  temperature  of  the  room.  The 
water  should  be  vigorously  stirred  during  the  condensation. 

The  calorimeter  is  then  removed  and  the  stirring  continued  until 
the  temperature  reaches  a  maximum,  when  the  reading  t,  is  taken 
and  recorded.  The  temperature  t2,  of  the  steam  entering  the  cal- 
orimeter 'is  to  be  determined  by  taking  the  barometric  reading  at 
the  time  of  the  experiment,  and  referring  to  Table  IX.  A  third 
weighing  determines  the  mass  of  steam  condensed,  M2. 

Since  M2  is  usually  a  small  mass  any  loss  of  water  due  to  drops 
adhering  to  the  exit  tube  from  the  water  trap-  leads  to  a  relatively 
large  error  in  the  mass  of  steam  condensed,  and  should  be  taken 
into  account  for  accurate  work.  If  steam  be  allowed  to  enter  the 
calorimeter  too  rapidly,  the  tube  leading  into  it  is  covered  on  its 
inner  surface  with  minute  drops  which  are  difficult  to  recover. 
Owing  to  the  large  influence  of  the  above  sources  of  error  the  fol- 
lowing method  is  to  be  preferred. 

Second  Method.  The  apparatus  (Fig.  47),  consists  of  a  closed 
copper  vessel  or  calorimeter,  provided  with  a  stirrer  and  an  open- 
ing for  a  thermometer ;  inside  the  closed  vessel  is  a  second  smaller 
vessel  into  which  steam  is  passed  and  there  condensed.  The 


CALORIMETRY 


123 


calorimeter  proper  having  been 
carefully  dried  and  weighed,  is 
filled  nearly  full  of  water  at  a 
temperature  some  15°  to  20°  be- 
low room  temperature  and  again 
weighed.  The  difference  is  M±. 
Steam  is  then  generated  in  a 
small  glass  retort  connected  with 
the  inner  vessel  into  which  the 
steam  passes  and  condensing 
gives  up  its  heat  to  the  calorime- 
ter and  its  contents.  The  tem- 
perature of  the  cold  water  tlt 
is  to  be  taken  just  before  the 
steam  enters  the  calorimeter. 

Steam  is  allowed  to  pass  in  until  the  resulting  temperature  rises 
as  much  above  room  temperature  as  the  initial  temperature  of  the 
water  was  below  it.  The  flame  is  then  removed  and  the  tempera- 
ture t,  carefully  determined.  The  amount  of  steam  condensed  is 
found  by  weighing  -the  small  retort  before  and  after  the  experi- 
ment. The  difference  is  the  mass  of  steam  condensed,  M2.  The 
temperature  t.,,  of  the  steam  entering  the  calorimeter  is  to  be  de- 
termined from  the  barometric  reading  as  before.  Care  must  be 
taken  to  prevent  radiation  from  the  retort  and  the  flame  under 
it  from  reaching  the  calorimeter.  Stirring  should  be  continued 
after  removal  of  the  flame  until  the  temperature  ceases  to  rise. 


Fig-  47- 


FORM    OF   RECORD. 

Exercise  44.     To  determine  the  heat  of  vaporization  of  water. 

Date  ................ 

Weight  of  calorimeter  =  ........ 

Specific  heat  of  calorimeter  =  ........ 

Water  equivalent  in  —  ........ 

Weight  of  calorimeter  with  water       =.  ........ 


Weight  of   retort  before  experiment   =  ........ 

Weight  of  retort  after  experiment       =  ........ 


-I/, 

tt 

1 

L 

124 


PHYSICAL    MEASUREMENTS 


95- 
Tin. 


-/  —  ~- 


Exercise  45.     Melting   Point  and   Heat  of  Fusion  of 

The  tin  is  contained  in  an  iron  vessel  I,  Fig.  48,  which  is 
closed  by  a  cover,  provided  with  a  slit 
5  to  admit  the  stirrer  S,  and  carrying  a 

narrow  tube  T,  to  receive  a  thermom- 
eter reading  to  360°  C,  or  a  thermo- 
element. The  tube  should  extend  well 
into  the  tin,  and  contain  a  small 
quantity  of  mercury  to  insure  good 
thermal  contact.  In  case  a  thermom- 
eter is  used  the  stem  correction  (Ar- 
ticle 81)  must  not  be  neglected.  It  is 
best  to  place  the  iron  vessel  inside  a 

larger  one,  or  surround  it  with  an  asbestos  screen  in  order  to 
avoid  irregularities  in  cooling  due  to  draughts  of  air. 

Heat  the  vessel  slowly  until  all  the  tin  is  fused  and  the  tempera- 
ture has  risen  to  about  280°  C.  Turn  out  the  flame  and  take 
temperature  readings  every  minute,  stirring  the  molten  metal 


Fig.  48. 


T, 


ttC 


D  t; 


Fig.  49. 


constantly  until  it  solidifies.  Continue  the  readings  until  the 
temperature  has  fallen  to  about  180°  C.  Plot  times  and  corrected 
temperatures.  The  curve  obtained  will  be  similar  to  that  shown 
in  Fig.  49.  A  study  of  the  curve  thus  found  will  show  the  fol- 
lowing facts : 

i.     Solidification  occurs  where  the  curve  becomes  practically 


CALORIMETRY  125 

horizontal.     The  mean  temperature  reading  for  this  part  of  the 
curve  is  the  melting  point  of  tin. 

2.  To  find  the  heat  of  fusion  of  tin,  determine  the  rate  of  cool- 
ing before  the  melting  point  is  reached  and  after  the  metal  is  all 
solidified.  These  rates  are  given  by  the  expressions  *  "~  and 

tz —  ti 

T\  ~  Tf"  where  T  denotes  temperature,  and  t  denotes  time. 

t  z —  t  i 

If  now  we  call  the  specific  heat  of  molten  tin  ^  =  0.064,  and 
that  of  solid  tin  s2  =  0.060,  and  if  M  represents  the  mass  of  tin, 
and  m  the  water-equivalent  of  the  iron  vessel,  stirrer,  etc.,  then  the 
quantity  of  heat  Qlt  lost  by  radiation  during  the  interval  t2 — flt  is, 
since  cw  equals  unity,  given  by  the  numerical  equality 

Q1=(Msl  +  m)  ( Ti  —  Tz)  ( 140) 

and  that  lost  during  the  interval  t'2  —  t\,  is 

0,=  (M  *  +  »»)(  TV -TV)        -  (141) 

and  for  R,  the  rate  at  which  heat  is  lost  we  have  the  two  values 


,      N       1— 8 

-f  m)  *-; : 

ti  — •  /i 

(142) 


Take  the  mean  of  these  two  values  for  R. 

Continue  the  straight  parts  of  the  curve  where  only  cooling 
occurs,  until  they  intersect  the  line  of  constant  temperature  in  the 
points  c  and  d.  We  may  then  assume  that  during  the  time 
t  =  C  D,  corresponding  to  the  era  of  constant  temperature  c  d, 
the  process  of  solidification  developed  sufficient  heat  to  supply  the 


126 


PHYSICAL,  MEASUREMENTS 


loss  due  to  radiation  and  thus  maintained  the  temperature  con- 
stant. This  quantity  of  heat  is  ML,  if  L  be  the  heat  of  fusion  of 
tin.  Again  the  quantity  of  heat  radiated,  during  this  interval  is 
Rt,  hence 

ML  =  Rt  (143) 

and 


Rt 


d44) 


FORM   OF   RECORD. 

Exercise   45.      To   determine   the   melting  point   and  heat   of 
fusion  of  tin. 

Date 

Weight  of  vessel  •=. 

Weight  of  tin  = 

Water  equivalent  ==. 


Time 


Temperature 


TV  — 7Y=... 


Fusing  point  of  tin  =  . . .  . 


L  =  .. 


VAPOR  PRESSURE. 

96.     Measurement  of  Vapor  Tension.     The  vapor  tension 

of  a  liquid  is  the  pressure,  measured  in  millimeters  of  mercury  at 
o°  C,  exerted  by  its  saturated  vapor  produced  by  evaporation  in 
a  vacuum.  An  increase 'in  temperature  produces  an  increase  in 
vapor  tension,  but  vapor  tension  increases  at  a  more  rapid  rate 
than  the  temperature.  When  the  pressure  of  the  saturated  vapor 
above  a  liquid  becomes  equal  to  the  atmospheric  pressure  exerted 
upon  its  surface,  the  liquid  begins  to  boil.  We  are  thus  in  position 
to  measure  vapor  tension  in  two  ways : 

(a)  By  measuring  the  depression  of  a  barometric  column  of 
mercury,  due  to  saturated  vapor  formed  in  a  vacuum. 

(b)  By  determining  the  boiling  point  under  different  pres- 
sures.   The  curve  showing  the  vapor  tension  as  a  function  of  the 
temperature,  is  called  the  vapor  tension  curve. 


VAPOR  PRESSURE 


127 


o 


97,  Exercise  46.  Vapor  Tension  of  Ether.  The  apparatus, 
Fig.  50,  consists  of  a  U-shaped  tube  with 
unequal  arms,  of  which  the  shorter,  about 
50  cms  in  length,  is  closed  and  the  other 
open.  The  short  arm  and  part  of  the 
longer  arm  of  the  tube  is  first  filled  with 
mercury,  the  mercury  being  boiled  in  the 
tube  to  expel  the  air  and  then  a  small 
quantity  of  air  free  ether  is  introduced 
into  the  shorter  arm. 

Insert  the  tube  into  a  tall  beaker  filled 
with  water  and  furnished  with  a  ther- 
mometer and  a  stirrer.  Vary  the  temper- 
ature of  the  water  between  10°  and  40°  C, 
and  measure  the  vapor  pressure  for  each 
temperature  with  a  cathetometer  or  with 
a  meter  stick.  Care  must  be  taken  not  to 
raise  the  temperature  of  the  water  at  any 
time  much  above  the  normal  boiling 
point  of  ether.  Enough  ether  should  be 
in  the  tube  so  that  at  any  temperature 
obtained  some  of  it  remains  in  liquid 
form.  Note  the  barometric  reading.  Plot  the  total  pressure  in 
cms  of  mercury  as  a  function  of  temperature  and  determine  from 
the  curve  the  boiling  point  for  a  pressure  of  760  mms  of  mercury. 

FORM    Of   RECORD. 

Exercise  46.    To  determine  the  vapor  tension  of  ether. 


Fig.  50. 


Temperature 


Reading  on 
short  arm 


On  long 


Date, 
Difference 


P  in  cms  of  Hg 


Normal  boiling  point 

98.  Exercise  47.  Vapor  Tension  of  Water  at  Various  Tem- 
peratures. As  pointed  out  in  the  previous  exercise,  a  liquid  be- 
gins to  boil  as  soon  as  its  vapor  tension  equals  the  pressure  upon 
the  liquid.  The  easiest  way  therefore  to  determine  the  vapor 
tension  of  a  liquid  at  temperatures  near  its  boiling  point,  is  to 


128 


PHYSICAL,    MEASUREMENTS 


vary  the  pressure  upon  the  liquid  and  determine  the  temperature 
at  .which  it  will  boil  under  the  new  pressure. 

The  apparatus  Fig.  51,  consists  of  a  metallic  flask  A,  contain- 
ing the  boiling  liquid,  and  furnished  with  a  tight  fitting  rubber 
stopper,  through  which  pass  a  thermometer  and  the  glass  tube 
leading  to  the  condenser  C.  A  second  flask  B  is  connected  to 
the  upper  end  of  the  condenser  and  to  a  stopcock  tube  to  which 


Fig.  51. 


the  small  mercury  manometer  M  is  fused.  This  tube  can  be 
closed  against  the  external  air  by  the  stopcock  V.  The  reading  of 
the  barometer  at  the  time  of  the  experiment  plus  or  minus  the 
difference  in  the  height  of  the  mercury  in  the  two  arms  of  the 
manometer  gives  the  pressure  under  which  the  liquid  boils,  and 
the  thermometer  gives  the  corresponding  temperature. 

A  simple  method  of  increasing  the  pressure  above  the  liquid 
is  to  start  with  the  stopcock  V  closed  when  the  flame  is  placed 
under  the  flask  A.  Soon  the  pressure  will  rise.  If  necessary  the 
pressure  may  be  reduced  by  small  steps  by  quickly  turning  the 
stopcock  around  180  degrees. 


VAPOR  PRESSURE 


129 


In  order  to  prevent  radiation  from  the  thermometer  bulb 
towards  the  cooler  outside  it  is  advisable  to  surround  it  by  a 
cylinder  of  asbestos. 

Vary  the  pressure  over  ranges  assigned  by  the  instructor,  and 
note  the  temperatures  at  which  boiling  takes  place  in  each  case. 
Put  glass  beads  in  the  flask  A,  to  avoid  bumping.  Plot  pressures 
and  temperatures. 

FORM    OF  RECORD. 

Exercise  47.  To  determine  the  vapor  tension  of  water  at 
temperatures  in  the  neighborhood  of  the  boiling  point. 


Date. 


Barometer  reading... 
Thermometer  No.  . .  . 


Temperature 


Manometer 
Tube  a  I  Tube  b 


CHAPTER  VIII. 
ELECTRICAL  MEASUREMENTS. 

UNITS  AND  STANDARDS. 

99.    Resistance.    The  practical  unit  of  resistance  is  the  ohm. 
It  is  represented  by  the  resistance  offered  to  an  unvarying  cur- 


Fig.  52. 

rent  by  a  column  of  mercury  at  the  temperature  of  melting  ice, 

14.4521  grams  in  mass,  of  a  constant  cross-sectional  area  and  of 

length  106.3  centimeters. 

For  practical  purposes  standard  ohms  and 
multiples  of  the  ohm  are  made  of  coils  of  wire, 
usually  of  manganin,  an  alloy  whose  resistance 
varies  but  little  with  the  temperature,  and 
which  has  a  small  thermo-electromotive  force 
against  copper,  mounted  in  suitable  protective 
cases.  The  makers  usually  furnish  certificates 
for  these  coils  issued  by  the  testing  laboratories 
of  the  different  countries. 
•  Fig.  52  represents  some  types  of  the  Ger- 
man or  Reichsanstalt  standards.  Their  re- 
sistance varies  slightly  with  humidity.  This 

has  led  to  the  construction  of  standards  in  hermetically  sealed 


Fig.  53. 


ELECTRICAL   UNITS  13! 

cases  filled  with  oil  which  has  been  carefully  freed  from  moisture 
and  air.  These  standards  were  proposed  by  the  Bureau  of  Stand- 
ards in  Washington  and  are  called  the  N.B.S.  standards  of  resist- 
ance, (Fig.  53). 

For  ordinary  measurements,  resistance  boxes  containing  a 
number  of  coils  of  insulated  wire,  wound  on  bobbins  non-induct- 
ively,1  are  in  general  use.  The  top  of  the  box  containing  the 
coils  is  usually  of  ebonite  and  car- 
ries on  its  upper  surface  a  number 
of  heavy  brass  blocks  so  arranged 
that  connection  between  adjacent 
blocks  may  be  made  by  means  of 
plugs  inserted  between  them.  The 
ends  of  the  separate  coils  (Figs. 
54  and  55),  are  fastened  to  the 
ends  of  adjacent  blocks,  so  that 
when. any  plug  is  removed  the  cur-  Fig.  54- 

rent  passes  from  one  block  to  the  next  by  passing  through  the 
connecting  coil.  In  this  way  any  resistance  may  be  added  by 
removing  the  proper  plugs. 

When  the  plug  is  inserted  the  current  passes  from  block  to 
block  through  the  plug  itself,  the  resistance  of  which  must  of 
course  be  negligible.  On  this  account  the  plugs  must  fit  accurately 
and  be  kept  bright  and  clean,  the  ebonite  must  at  all  times  be  kept 
free  from  dust  or  moisture  and  never  be  allowed  to  stand  in  the 
sun,  as  the  ebonite  disintegrates  slowly  under  the  action  of  sun- 
light, a  conducting  layer  of  sulphur  forms  on  the  surface,  and  the 
efficiency  of  the  coils  is  impaired.  The  plugs  should  at  all  times 
be  handled  by  their  hard  rubber  tops,  they  should  be  inserted 
with  a  gentle  twisting  motion,  and  should  all  be  loosened  before 
the  box  is  returned  after  use. 


*Two  methods  of  non-inductive  winding  are  used;  in  the  one  case  the 
required  lengjh  of  wire  is  measured  off,  doubled  upon  itself  and  then 
wound  so  doubled,  upon  the  spool.  In  the  second  method  the  wire  is 
wound  in  one  direction  only  in  each  layer,  but  in  opposite  direction  in 
consecutive  layers.  By  this  method  the  capacity  of  the  coil,  which  by  the 
first  method  may  be  quite  appreciable,  especially  in  coils  of  high  resistance, 
is  much  reduced.  (Chaperon.) 


132 


PHYSICAL   MEASUREMENTS 


In  the  common  form  of  resistance  boxes  there  are  only  four 
coils  for  each  decade,  namely  of  I,  2,  3  and  4  units,  or  I,  i,  2 
and  5  units  (Fig.  86). 


Fig.  55- 

In  the  "decade"  resistance  boxes  the  inconvenience  of  adding 
up  the  resistances  of  the  separate  coils  has  been  avoided,  see  Fig. 


Fig-  56.  Fig.  57. 

56.  In  each  row  of  blocks  representing  a  decade  a  plug  is  placed 
between  two  blocks  and  inserts  in  the  circuit  as  many  units  as 
are  indicated  by  the  number  on  one  of  the  blocks. 


UNITS  133 

An  ingenious  method  in  which  only  four  coils  are  used  foi 
each  decade  has  been  devised  by  Leeds  and  Northrup.  The  ar- 
rangement is  shown  by  Fig.  57.  In  each  decade  there  is  one  re- 
sistance coil  of  i  unit,  one  of  2,  and  two  coils  of  3  units ;  for 
example  if  a  plug  be  inserted  between  the  two  blocks,  marked 
"4"  the  current  must  pass  through  resistances  I  and  '3',  while  2 
and  3  areshortcircuited.  These 
boxes  combine  the  advantage 
of  the  decade  plan  with  cheap- 
ness of  construction. 

In  the  "dial"  resistance 
boxes,  built  on  the  decade 
plan,  the  blocks  are  arranged 
in  a  circle  and  contact  with  a 
central  ring  is  made  by  a  slid- 
ing contact  rotating  around 
the  center  of  the  circle,  (see 
Fig.  58).  F^  58. 

Previous  to  the  adoption  of  the  ohm  various  other  units  of  re- 
sistance were  employed  and  resistance  boxes  representing  these 
earlier  standards  are  sometimes  met  in  practice.  The  most  com- 
mon are  the  British  Association  Unit,  proposed  by  the  British 
Association  in  1864,  and  the  legal  ohm  adopted  at  the  Paris  Con- 
gress in  1884.  The  relation  between  the  ohm  and  these  units  is 
as  follows: 

i  ohm  =  1.01358  B.  A.  units  =  1.0028  legal  ohms, 
i  B.  A.  unit  =  0.9866  ohm. 
i  legal  ohm  =  0.9972  ohm. 

In  reporting  work  in  measurements  of  resistance  the  student 
should  state  explicitly  which  units  have  been  used. 

The  term  rheostat  is  used  for  an  unknown  resistance  of  either 
fixed  or  variable  value,  and  employed  in  work  with  currents  ex- 
ceeding o.i  ampere. 

The  ends  of  the  lead  wires  connecting  the  different  instruments 
should  be  bright  and  clean  and  clamped  firmly  by  the  binding 
posts,  since  loose  connections  offer  high  and  variable  resistance. 
It  is  a  bad  practice  to  connect  two  wires  by  twisting  them  together. 


134 


PHYSICAL    MEASUREMENTS 


100.  Current.     The  practical  unit  of  current  is  the  ampere. 
The  ampere  is  represented  sufficiently  well  for  practical  purposes 
by  "the  unvarying  current  which  will  deposit  silver  from  silver 
nitrate  at  the  rate  of  0.001118  grains  per  second." 

101.  Electromotive  force.     The  practical  unit  of  electromo- 
tive force  is  the  volt.    The  volt  is  the  electromotive  force  which 
steadily  applied  to  a  conductor  whose  resistance  is  one  ohm  will 
produce  a  current  of  one  ampere.     The  Cadmium  cell  is  chiefly 
used  as  the  practical  standard  of  electromotive  force.  This  cell  has 
for  its  positive  electrode  mercury  covered  by  a  paste  of  mercurous 
sulphate  and  a  saturated  solution  of  cadmium  sulphate,  and  for 
the  negative  electrode  cadmium  amalgam,  placed  in  the  other  leg 
of  the  H-shaped  vessel,   (Fig.  59),  and  covered  by  a  saturated 


CdSO4  Crystals 
Hg2S04 


CdSO4  Crystals 
Cd  Amalgam 


Fig-  59- 

solution  of  cadmium  sulphate.  Saturation  of  the  cadmium  sulphate 
solution  is  secured  by  an  excess  of  cadmium  sulphate  crystals. 
The  glass  tubes  are  hermetically  sealed.  The  E.  M.  F.  of  this 
cell  at  the  temperature  t°C  is  given  by  the  formula 

Et=  1.0183  —  0.00004  (t  —  20)  volts.  (i45) 

The  legal  form  of  this  cell  is  not  portable  and  suffers  the  addi- 
tional disadvantage  of  a  slight  lag  in  the  E.  M.  F.,  since  the  dens- 
ity of  the  solution  adjusts  itself  to  a  new  temperature  but  slowly, 
some  time  being  required  for  the  cadmium  sulphate  crystals  to 
dissolve  or  to  crystallize  out.  This  cell  should  never  be  placed  on 
a  short  circuit.  (Why?) 

Some  older  forms  of  standard  cells  are  still  in  frequent  use. 


ELECTRICAL   UNITS  135 

The  Clark  cell,  the  first  efficient  form  of  standard  cell,  differs 
from  the  cadmium  standard  cell  by  the  use  of  zinc  and  zinc  sul- 
phate in  place  of  cadmium  and  cadmium  sulphate.  In  other 
respects  its  construction  is  identical  with  that  of  the  cadmium 
cell.  The  E.  M.  F.  of  the  Clark  cell  is  given  by  the  formula 

Ht=  1433  —  0.00119  (t—  15)  volts.  (146) 

The  Carhart-Clark  cell  is  an  unsaturated  form  of  Clark  cell 
and  its  E.  M.  F.  is  about 

Et  =  1.44  —  0.00056  (t—  15)  volts.  (147) 

The  Western  Company  manufactures  portable  unsaturated 
cadmium  cells,  which  are  extremely  convenient  on  account  of  their 
negligible  temperature  coefficient.  Their  E.  M.  F.  is  very  con- 
stant and  about  1.019  volts;  the  exact  value  being  readily  deter- 
mined by  comparison  with  a  primary  standard. 

Standard  cells  are  used  in  all  cases  where  it  is  necessary  to 
determine  the  absolute  value  of  an  E.  M.  F.  or  of  a  'potential 
difference.  In  many  cases  however,  we  wish  simply  to  compare 
deflections  of  a  galvanometer.  In  others  the  conditions  may  be 
so  chosen  that  no  current  passes  through  the  galvanometer.  Such 
a  method  is  termed  a  zero  method.  In  experiments  employing  the 
zero  method  the  Leclanche  battery  may  be  used.  Where  steady 
deflections  are  required  however,  a  cell  of  constant  electromo- 
tive force  is  necessary.  In  such  cases  we  may  use  either  a  Daniell 
cell  or  a  storage  battery. 

To  set  up  a  Daniell  cell  (Fig.  60),  first  fill  the  porous  cup  con- 
taining the  amalgamated  zinc,  or  negative 
electrode,  two-thirds  full  of  zinc  sulphate 
solution  and  wait  until  the  solution  begins 
to  moisten  the  outside  of  the  cup,  before 
placing  the  cup  in  the  glass  jar  containing 
the  copper,  or  positive  electrode,  and  the 
copper  sulphate  solution.  In  order  to  avoid 
changes  in  the  internal  resistance  during  the 
experiment  it  is  well  to  short  circuit  the  cell 
for  ten  minutes  before  using.  Fl£-  6o- 

After  use  the  cell  must  be  taken  apart,  the  zinc  sulphate  poured 


136  PHYSICAL    MEASUREMENTS 

back  into  its  bottle,  the  porous  cup  thoroughly  washed  and  the 
zinc  rubbed  clean.  Copper  oxide  is  usually  deposited  on  the  zinc 
as  a  black  film.  This  may  be  readily  rubbed  off  while  it  is  moist, 
but.  if  it  be  allowed  to  dry  it  adheres  firmly  and  renders  it  difficult 
to  amalgamate  the  zinc  again.  The  E.  M.  F.  of  a  Daniell  cell  is 
about  i.i  volts  and  this  value  may  be  used  in  all  cases  where 
great  accuracy  is  not  required. 

When  a  constant  E.  M.  F.  of  more  than  one  volt  is  required,  a 
storage  battery  (E  =  2.2  volts)  may  be  employed.  On  account 
of  the  very  low  internal  resistance  of  a  storage  battery  great  care 
must  be  exercised  to  avoid  short  circuiting  the  cell.  The  student 
using  a  storage  battery  should  always  leave  one  of  the  electrodes 
disconnected  until  the  instructor  has  seen  and  approved  of  the 
arrangement  of  the  apparatus. 

1  02.  Quantity  of  electricity.  The  practical  unit  of  quantity 
is  the  coulomb.  The  coulomb  is  the  quantity  of  electricity  trans- 
ferred by  a  current  of  one  ampere  in  one  second. 

103.  Capacity.  The  practical  unit  of  capacity  is  the  farad. 
The  farad  is  the  capacity  of  a  condenser  which  is  charged  to  a 
potential  of  one  volt  by  a  quantity  of  one  coulomb.  The  micro- 
farad =  icr6  farad,  is  commonly  used  as  the  measure  of  capacity. 
Secondary  standards  of  capacity  are  made  in  the  form  of  con- 
densers with  solid  dielectrics.  A  large  number  of  sheets  of  tin- 
foil interleaved  with  mica  or  paraffined  paper  are  placed  in  a  bath 
of  melted  paraffin  in  a  vacuum  chamber,  to  remove  air  bubbles, 
and  allowed  to  cool.  Alternate  sheets  of  tin-foil  are  then  con- 
nected and  joined  to  one  binding  post,  the  remainder  to  another, 
thus  forming  the  two  ends  or  terminals  of  the  condenser.  A  sub- 
divided condenser  is  made  by  connecting  all  the  leaves  of  one  set 

t0  °ne  bar>  Earth,   (Fi£  6l)' 


- 

as  a  terminal,  and  dividing  the 

leaves   of   the   other   set   into 
some  number  of  divisions  each 
of  which  is  connected  to  a  sep- 
Fig.  61.  arate  bar  which  in  turn  may  be 


ELECTRICAL,   UNITS 


137 


connected  to  a  second  binding  post  by  means  of  a  plug.     When 
any  division  is  to  be  used  the  plug  is  inserted  for  that  division. 
In  another  form,  made  by  Leeds  and  Northrup,  (Fig.  62),  the 
subdivisions  of  the  condenser 
are   placed   between   two   ad- 
joining crossbars,  thus  allow- 
ing their  use  in  series  as  well 
as   in  parallel.     Where  must 
the  plugs  be  inserted  to  obtain 
I  mf.,  0.4  mf.  or  0.25  mf  ? 


Fig.  62. 


104.  Selfinductance.  The  practical  unit  of  selfinductance 
is  the  henry.  A  henry  is  the  selfinductance  in  the  circuit  when 
the  E.  M.  F.  induced  in  the  circuit  is  one  volt,  while  the  inducing 
current  varies  at  the  rate  of  one  ampere  per  second. 

The  usual  form  of  selfin- 
ductance (Fig.  63) ),  consists  of 
two  coils  in  series,  one  fixed 
and  the  other  movable  about  a 
diameter  of  the  fixed  coil  as  an 
axis.  The  movable  coil  may  be 
rotated  through  180°  and  in 
this  way  the  selfinductance  may 
be  varied  considerably.  The 
instrument  is  calibrated  empir- 
ically. The  coils  are  wound  on 
wood,  and  metal  is,  so  far  as 
possible,  entirely  avoided  in  the 
pjg  63  construction. 

Other  standards  of  inductance  (Fig.  64),  usually  of  single 
value  only,  are  wound  upon  spools  of  non- 
magnetic and  nonconducting  material,  such 
as  mahogany,  marble  or  soapstone.  The 
last  is,  however,  generally  magnetic  in  slight 
degree,  and  standards  wound  upon  soap- 
stone  should  be  mistrusted. 
Fig.  64. 


138  PHYSICAL    MEASUREMENTS 

INSTRUMENTS. 

105.  Keys.  In  most  forms  of  apparatus  for  making  electrical 
measurements,  it  is  necessary  to  allow  the  current  to  pass  through 
the  measuring  instrument  for  but  a  short  space  of  time.  Keys  are 
provided  for  the  purpose  of  closing  and  opening  the  circuit  as 
may  be  desired.  The  ordinary  key  is  so  arranged  that  when 
pressed  down,  contact  is  made  between  two  platinum  points  and 
the  circuit  is  closed.  On  releasing  the  key  the  circuit  is  opened. 
For  closed  circuit  work,  plug  keys  or  knife  switches  of  various 
forms  are  used. 

In  work  with  the  Wheatstone's  bridge  it  is  necessary  to  close 
the  circuit  through  the  galvanometer,  after  the  current  in  the 
arms  of  the  bridge  has  reached  a  constant  value.  For  this  pur- 
pose a  double  key  (Fig.  65)  is  provided,  in  which  the  contacts 
for  the  circuit  through  the  bridge  arms,  and  that  through  the  gal- 
vanometer, are  made  successively  in  the  order  mentioned.  Such 
a  key  is  termed  a  successive  contact  key. 


Fig.  65.  Fig.  66. 

It  is  frequently  necessary  to  reverse  the  direction  of  the  current 
through  an  instrument  without  loss  of  time.  This  is  most  con- 
veniently effected  by  means  of  the  Pohl's  commutator  (Fig.  66). 
This  consists  of  four  cups  containing  mercury  connected  by  cross 
wires  as  indicated  in  the  figure,  and  a  light  frame  of  wires  by 
which  two  other  cups  at  the  end  of  the  block  may  be  put  in  con- 
nection with  the  pair  of  cups  on  either  side.  If  the  source  of 
current  be  connected  to  the  pair  of  binding  posts  at  the  ends  of 
the  block,  and  the  galvanometer  to  the  two  binding  posts  on  either 


GALVANOMETERS  139 

side,  the  current  is  passed  through  the  instrument  in  one  direction 
or  the  other  by  tilting  the  frame  from  side  to  side. 

All  mercury  contacts  should  be  kept  clean  and  should  have  the 
ends  of  the  metal  dipping  into  the  mercury  well  amalgamated. 

1 06.  Galvanometers.  The  space  about  a  magnet  or  about  a 
wire  carrying  a  current  of  electricity  is  called  a  magnetic  field. 
Such  a  field  is  conceived  to  be  filled  with  lines  of  magnetic  induc- 
tion. The  direction  of  these  lines  is  assumed  to  be  the  direction 
along  which  a  free  north  seeking  pole  would  tend  to  move.  The 
lines  of  magnetic  induction  are  said  to  run  out  from  the  north 
pole  of  a  magnet,  curve  round  through  the  air  and  re-enter  the 
magnet  at  the  south  pole.  In  the  case  of  a  wire  carrying  an  elec- 
trical current  the  lines  of  induction  are  concentric  circles  sur- 
rounding the  wire. 

Whenever  two  magnetic  fields  are  brought  near  to  each  other 
there  arises  a  stress  between  them,  tending  to  turn  the  fields  into 
such  a  position  that  they  will  mutually  include  the  greatest  num- 
ber of  lines  of  induction.  Obviously  the  moment  of  this  stress  will 
be  greatest  when  the  two  systems  of  lines  of  induction  stand  at 
right  angles  to  each  other.  This  is  the  position  adopted  in  gal- 
vanometers and  electro-dynamometers.  In  instruments  de- 
signed to  measure  currents,  at  least  one  of  the  magnetic  fields 
must  arise  from  the  current  flowing  through  a  coil  of  wire  and 
hence  must  vary  as  the  strength  of  the  current.1  The  other 
magnetic  field  may  be  produced  by  a  .permanent  magnet  as  in  the 
galvanometer,  or  by  a  second  coil  carrying  a  current,  as  in  the 
electro-dynamometer.  One  of  the  magnetic  fields  must  be  capa- 
ble of  rotation.  Galvanometers  may  be  divided  into  two  classes  :2 

(a)  Galvanometers  with  stationary  coil  and  movable  system 
of  magnets;  needle  type;  (Fig.  67). 


1  College  Physics,  Article  257. 

-  College  Physics,  Article  260  and  261. 


140  PHYSICAL    MEASUREMENTS 

(b)     Galvanometers    with    stationary    magnets 
and  movable  coil;  d'Arsonval  type.     (Fig.  68.) 

In  all  measuring  instruments  the  deflecting  mo- 
ment due  to  the  current  to  be  measured,  must  be 
balanced  against  a  restoring  or  directing  moment 
which  tends  to  restore  the  system  to  its  original 
position.  When  the  system  thus  subjected  to  the 
action  of  two  moments  comes  to  rest  we  know 
that  the  moments  of  the  deflecting  and  restoring 
forces  are  equal  and  opposite  in  direction.  The 
angle  of  deflection  is  determined  in  one  of  several 
ways  and  the  current  determined  as  a  function  of 
this  angle.  The  directing  moment  may  be  due  to 
the  action  of  an  independent  magnetic  field  upon 
the  movable  magnetic  needle,  or  to  the  torsional 
Fig.  67.  moment  of  the  suspending  wire. 

The  sensitiveness  of  a  galvanometer  may  be  increased  by  de- 
creasing the  directing  or  restoring  moment. 

107.  The  Astatic  Galvanometer.     In  the  astatic  galvanom- 
eter the  moving  system  is  composed  of  two  magnets  or  systems 
of  magnets  of  nearly  equal  strength  placed  one  above  the  other 
with  poles  opposed.     One  of  the  needles  is  placed  within  a  coil, 
the  other  needle  either  outside,  or,  better  still,  enclosed  in  a  second 
coil  through  which  the  current  flows  in  the  opposite  direction.    In 
this  way  the  magnetic  moment  of  the  system  with  respect  to  the 
earth's  field  is  greatly  reduced. 

On  the  other  hand  a  magnet  placed  under  or  over  a  movable 
magnetic  needle  may  be  made  to  exert  any  desired  directive  mo- 
ment. Such  a  magnet  is  termed  a  controlling  magnet.  Some 
very  sensitive  galvanometers  employ  both  an  astatic  system  and 
a  controlling  magnet,  the  latter  being  placed  in  such  a  direction 
as  to  oppose  the  magnetic  field  of  the  earth. 

1 08,  The   d'Arsonval   Galvanometer.     The  d'Arsonval  gal- 
vanometer, (Fig.  68),  is  more  convenient  for  ordinary  measure- 
ments.    The  coil   (Fig.  69)     is  suspended  by  means  of  a  fine 
metal  wire  or  ribbon,  and  is  attached  to  the  base  of  the  instru- 


GALVANOMETERS 


141 


ment  by  another  wire  or  metallic  spiral  spring.  The  current  en- 
ters and  leaves  the  coil  by  these  upper  and  lower  suspensions. 
The  magnetic  field  of  the  permanent  magnet  is  directed  from  N 
to  S  across  the  gap  in  which  the  coil  is  suspended.  The  current 
flowing  through  the  coil  sets  up  a  magnetic  field  whose  direction 
may  be  found  by  the  right  hand  rule1,  and  is  indicated  in  the  fig- 
ure by  the  crosses  X,  where  the  lines  of  induction  enter  the  plane 
of  the  paper  and  by  the  dots  • ,  where  the  lines  come  out.  The 


Fig.  68. 


Fig.  69. 


coil  tends  to  place  itself  so  that  its  lines  of  induction  inside  the 
coil  are  parallel  to  those  of. the  field  of  the  stationary  magnet. 
Thus  with  the  current  flowing  as  indicated,  the  side  ab  tends  to- 
rise  out  of  the  paper.  The  coil  comes  to  rest  when  the  deflecting 
moment,  due  to  the  action  of  the  fields,  equals  the  torque  iir  the 
suspension.  The  finer  the  suspension,  the  more  sensitive  is  the 
instrument.  Deflections  are  usually  observed  by  means  of  mirror 
and  scale.  The  d'Arsonval  galvanometer  has  the  great  advan- 
tage that  on  closed  circuit  considerable  damping  effect  is  pro- 
duced by  electro-magnetic  induction  in  the  moving  coil.  Why 
is  this?  The  degree  of  damping  depends  largely  upon  the  resist- 


1  College  Physics,  Article  256. 


142  PHYSICAL    MEASUREMENTS 

ance  of  the  circuit,  being  larger,  the  smaller  the  resistance.  For 
each  galvanometer  a  resistance  of  the  circuit  can  be  found,  for 
which  the  swing  of  the  coil  changes  from  an  oscillatory  to  an 
aperiodic  motion.  This  limiting  resistance  is  called  the  critical 
resistance  of  the  circuit,  and  it  can  be  shown  that  with  this  crit- 
ical resistance  the  action  of  the  galvanometer  is  quickest. 

The  instrument  is  also  practically  independent  of  the  surround- 
ing magnetic  field  and  is  consequently  free  from  disturbances 
arising  from  variations  in  the  intensity  of  this  field,  which  often 
prove  very  troublesome  in  galvanometers  of  the  needle  type.  The 
d'Arsonval  galvanometer  has  the  additional  advantage  that  it  may 
be  placed  in  any  position,  while  instruments  of  the  needle  type 
must  be  set  so  as  to  have  the  plane  of  the  coils  in  the  magnetic 
meridian.  The  needle  type  however,  has  usually  greater  sensitive- 
ness, although  the  d'Arsonval  type  is  sufficiently  sensitive  for 
most  purposes. 

109.  Methods  of  Observation.  In  some  instruments  the  mov- 
ing system  is  furnished  with  a  light  pointer  playing  over  a  scale, 
but  in  the  more  sensitive  galvanometers  the  deflections  are  ob- 
served by  means  of  a  mirror  attached  to  the  moving  system.  This 
mirror  may  be  either  concave  or  plane.  In  the  first  case  an  illum- 
inated slit  is  focused  by  the  mirror  upon  a  semi-transparent  scale 
and  the  deflections  are  read  directly  from  the  scale.  In  the  sec- 
ond case  a  telescope  is  employed  to  view  the  image  of  a  scale 
reflected  in  a  plane  mirror  (see  Article  29).  Both  methods  are 
in  common  use. 

no.  Shunts.  It  frequently  happens  that  the  current  to  be 
measured  will  produce  a  deflection  too  great  to  be  observed,  or  in 
some  cases  it  may  even  endanger  the  instrument  itself.  In  such 

cases  we  may  reduce  the  current 
flowing  through  the  galvanom- 
eter by  means  of  a  resistance 
connected  in  parallel  with  it. 
This  resistance  is  called  a  shunt. 
Let  g  be  the  resistance  of  the 
galvanometer,  s  that  of  the  shunt, 
then  the  resistance  of  the  two 


GALVANOMETERS  143 

circuits  in  parallel  is         gs    ,  and  by  Ohm's  law,1  if  /  denote  the 

9  ~T~  S 

total  current  and  7g  the  current  through  the  galvanometer,  we  have 


or 


If  we  observe  the  current  Ig  in  the  galvanometer,  then  the  total 

current  is  Ie.  9  "*"  s  ;  where  the  factor  g  +  s  is  called  the  multiply- 
s  s 

ing  power  of  the  shunt.    Let  the  galvanometer  resistance  be  n 
times  that  of  the  shunt,  (g==ns),  then  the  multiplying  power  is 

—  .  If  n  be  9,  99  or  999,  the  corresponding  values  of  the 
multiplying  power  are  10,  100,  and  1000.  The  makers  frequently 
furnish  with  the  galvanometer  shunt-boxes  containing  resistances 
equal  to  1/9,  1/99,  and  1/999  of  that  of  the  galvanometer,  in 
which  case  the  observed  current  7g,  is  equal  to  o.i,  o.oi,  or  o.ooi  /. 

in.  Exercise  48.  Calibration  of  a  Galvanometer  by  Ohm's 
Law.  The  object  of  this  experiment  is  to  determine  whether  the 
deflections  of  a  galvanometer  are  proportional  to  the  current 
flowing  through  it,  or  if  that  is  not  the  case,  to  ascertain  how  the 
deflections  vary  with  the  current.  To  obtain  currents  which 
stand  in  a  definite  relation  to  each  other  we  apply  Ohm's  law,  by 
connecting  the  terminals  of  the  galvanometer  to  different  points 
of  a  circuit  through  which  a  constant  current  is  flowing.  Then 
the  potential  difference,  P.  D.,  between  the  extremities  of  a  re- 
sistance r,  through  which  a  current  i  is  flowing,  is  equal  to  ir, 
and  this  potential  difference  causes  the  current  through  the  gal- 
vanometer producing  the  observed  deflection. 

The  simplest  arrangement  is  to  use  a  battery  of  constant  E.  M. 
F.2  and  to  send  the  current  through  a  straight  wire,  as  for  ex- 


*It  should  be  kept  in  mind  that  Ohm's  law  applies  to  constant  cur- 
rents only. 

2  The  student  may  observe  as  a  general  rule,  that  a  battery  of  constant 
E.  M.  F.  is  always  to  be  used  where  constant  deflections  are  to  be  obtained 
while  ordinary  cells  may  be  employed  for  zero  methods  or  ballistic 
methods. 


144 


PHYSICAL    MEASUREMENTS 


Q 


ample,  the  wire  of  a  slide  wire  bridge,  (Fig.  71),  and  connect  the 
galvanometer  to  two  points,  P  and  A,  on  this  wire,  where  A 
denotes  the  position  of  the  contact  maker.  Since  the  current 
through  the  wire  must  remain  constant,  no  matter  where  the  gal- 
vanometer is  attached,  it  is  evident  that  the  galvanometer  should 
have  a  very  high  resistance  as  compared  with  that  of  the  wire. 
Explain  this.  The  resistance  of  the  wire  of  a  slide  wire  bridge  is 
usually  about  0.2  of  an  ohm.  If  the  galvanometer  resistance  is 

less  than  2000  ohms, 
a  resistance  box  R2, 
with  sufficient  resist- 
ance to  bring  the  re- 
sistance of  the  galva- 
nometer up  to  2000 
ohms,  should  be  put  in 
series  with  it.  The  re- 
sistance box  R!  is  inserted  in  the  battery  circuit  in  order  to  reduce 
the  potential  difference  between  P  and  A  to  a  value  such  that 
with  the  maximum  length  of  wire  used  in  the  experiment  the 
deflections  of  the  galvanometer  will  still  be  on  the  scale. 

Move  the  point  A,  an  ordinary  contact  maker,  along  the  wire 
by  steps  of  5  cms  ,  from  5  to  95  cms  on  the  scale  and  observe  the 
successive  deflections  of  the  galvanometer.  Next  reverse  the  bat- 
tery current  and  repeat  the  observations.  The  mean  of  the  de- 
flections and  the  corresponding  lengths  between  P  and  A  are 
plotted.  The  resulting  curve  will  be  a  straight  line,  if  the  deflec- 
tions of  the  galvanometer  are  proportional  to  the  current  flowing 
through  it.  What  principle  besides  Ohm's  law  has  been  applied 
in  this  experiment? 

FORM    OF   RECORD. 

Exercise  48.  To  calibrate  a  galvanometer  by  means  of  Ohm's 
law. 

Galvanometer  No Date 

Resistance  boxes       Ri R« 

Length  Deflection  a  Deflection  b     •     Mean  Deflection 


112.     Exercise  49.    Figure  of  Merit  of  a  Galvanometer.  The 
figure  of  merit  /,  of  a  galvanometer  is  the  ratio  of  the  current 


GALVANOMETERS 


145 


to  the  deflection  which  it  produces  and  is  measured  by  that  cur- 
rent which  will  produce  a  deflection 
of  one  scale  division.  In  case  the 
scale  is  movable  the  distance  of  the 
scale  from  the  mirror  must  be  speci- 
fied, or,  still  better,  the  result  should 
be  calculated  for  a  distance  of  100 
cms.  This  exercise  furnishes  a  sim- 
ple application  of  Ohm's  law. 

The  exercise  may  be  performed  in 
either  of  the  following  ways : 

(a)     The  arrangement  is  as  shown  Fig.  72. 

in  Fig.  72.  Let  B  denote  a  battery  of  constant  electromotive 
force  E;  R  a  very  high  resistance;  g  and  b  the  resistances  of  the 
galvanometer  and  battery  .respectively  ;  then 


1  = 


if  no  shunt  is  needed,  and 


R+9+b   ' 
£ 


(150) 
(151) 


if  a  shunt  is  used. 

If  the  galvanometer  show  a  deflection  d,  on  the  passage  of  a 
current  IgJ  through  it,  we  may  assume,  in  the  great  majority  of 
cases,  that  this  deflection  is  proportional  to  the  current,  and  we 
have  the  relation 

fd  =  I  '  (152) 

The  factor  /  is  the  figure  of  merit  of  the  galvanometer. 
Hence  in  the  first  case 


d 


and  in  the  second  case 
f=^-  =  —         s 


(153) 


d54) 


Usually  b  as  well  as  g,  and  still  more    — £L-  ,  are  negligible  in 
comparison  with  R,  and  the  formulae  become  much  simpler. 


146 


PHYSICAL    MEASUREMENTS 


(&)     For  sensitive  galvanometers  which  have  no  permanent 
shunt  the  following  method  is  convenient:   Let  B,  (Fig.  73),  be 

a  cell  of  constant  electromotive  force. 
E,  and  close  the  circuit  through  P  and 
Q.  Then  take  the  potential  difference 
over  Q  to  produce  the  deflection  of  the 
galvanometer.  Q  must  be  very  small 
in  comparison  with  R.  Now  the  ap- 
plied potential  difference  is  — ^—  E 
and  our  formula  for  /  becomes 


Fig.  73- 


d55) 


P  +  Q 


In  practice  vary  the  resistance  R  between  150000  and  250000 
ohms  and  observe  the  deflections;  in  case  (b)  the  ratio 
may  also  be  varied. 

The  term  sensitiveness  of  a  galvanometer  is  frequently  used  in 
a  different  sense.  It  may  be  defined  as  the  resistance  which  the 
circuit  must  have  in  order  that  one  volt  may  produce  unit  deflec- 
tion. This  is  termed  the  ohm  sensitiveness  of  the  galvanometer, 


(156) 


The  potential  difference  per  unit  deflection  is  called 
the  volt  sensitiveness  of  the  galvanometer,  a  factor  of  the  high- 
est importance  in  most  electrical  measurements,  as  for  example  in 
work  with  the  Wheatstone  bridge,  the  potentiometer  or  the 
thermal  couple. 

If  in  Fig.  73  the  electromotive  force  of  the  battery  £  be 
known,  then  the  volt  sensitiveness  of  the  galvanometer  can  be 
shown  directly  to  be 

(157) 


In  order  to  give  a  definite  meaning  to  this  term,  the  resistance 


GALVANOMETERS 


147 


in  the  galvanometer  circuit  (d'Arsonval  galvanometer)  should  be 
the  critical  resistance.  Many  makers  give  the  sensitiveness  at 
the  terminals  of  the  galvanometer,  but  this  is  too  indefinite  since 
frequently,  owing  to  excessive  damping,  the  galvanometer  can- 
not be  used  upon  short  circuit. 


FORM    OF  RECORD. 


'Exercise  40.    To  determine  the  figure  of  merit  of  a  galvanom- 
eter. 


Galvanometer  No g  = 

Distance  of  mirror  from  scale 


Date. 


Q 

E 

R 

Q 

P 

P  +  Q 

d 

f 

f(R  +  9) 

113.  Ballistic  Galvanometers.  While  in  ordinary  galvanom- 
eters it  is  required  to  observe  deflections  due  to  a  steady  current 
flowing  through  the  instrument,  or  to  prove  the  absence  of  a  cur- 
rent from  the  absence  of  a  deflection,  it  is  often  necessary  to 
measure  the  quantity  of  electricity  passing  through  the  galvanom- 
eter, as  in  measuring  the  quantity  of  electricity  stored  in  a  con- 
denser. 

When  a  condenser  is  discharged  through  a  galvanometer  the 
current  rises  rapidly  to  a  maximum,  and  then  decreases  to  zero. 
In  such  a  case  a  galvanometer  having  a  coil  with  a  large  moment 
of  inertia  must  be  employed.  Such  an  instrument  is  termed  a  bal- 
listic galvanometer  and  its  advantage  consists  in  this,  that  the  coil 
remains  practically  at  rest  until  the  entire  quantity  of  electricity 
has  passed  through  it.  In  this  way  the  full  force  of  the  magnetic 
thrust  is  effective  in  starting  the  coil  which  moves  off  as  if  started 
by  a  blow.  For  small  angular  deflections  the  quantity  of  electric- 
ity may  be  set  proportional  to  the  deflection.1  Here  we  observe 
the  maximum  deflection  attained  by  the  system  on  the  first  throw 
of  the  needle,  and  not  a  constant  deflection.  The  quantity  of 


1  Carhart  and  Patterson,  Electrical  Measurements,  pp.  207-213. 


I48 


PHYSICAL    MEASUREMENTS 


electricity  per  unit  deflection  is  called  the  constant  of  the  ballistic 
galvanometer,  or  if  the  quantity  Q  gives  a  deflection  d,  then 

c=  &  (158) 

114.  Constant  of  Ballistic  Galvanometer.     The  constant  of 
a  ballistic  galvanometer,  especially  in  instruments  of  the  d'Ar- 
sonval  type,  depends  largely  upon  the  resistance  of  the  galvan- 
ometer circuit.     In  the  following  exercise  it  is  determined  with 
the  galvanometer  on  open  circuit,  in  which  case  it  matters  not 
how  much  resistance  we  add  to  the  galvanometer.     However  the 
value  thus  found  cannot  be  applied  to  the  instrument  when  used 
upon  a  closed  circuit,  without  taking  into  account  the  damping 
effect.     This  must  be  remembered  in  Exercises  74,  78  and  79. 

115.  Exercise  50.    Determination  of  the  Constant  of  a  Bal- 
listic Galvanometer.    To  determine  the  quantity  Q  giving  a  cer- 
tain deflection  d,  use  a  battery  of  known  electromotive  force  E, 

as  a  standard  cell,  and  charge  a  condenser 
of  known  capacity  by  depressing  the  key  to 
the  point  b  (Fig.  74).  Then  by  releasing 
the  key,  the  condenser  is  disconnected  from 
the  battery  and  discharged  through  the 
galvanometer.  Special  keys,  used  for  such 
experiments,  known  as  charge  and  dis- 
charge keys,  are  shown  in  Fig.  75. 
Fig.  74.  Letting  c  represent  the  constant  of  the 

ballistic  galvanometer,  Q  the  quantity  discharged,  and  d  the  de- 
flection, then 

s-\  r*  /-• 

(159) 


Fig.  75- 


GALVANOMETERS 


149 


where  C  is  the  capacity  of  the  condenser  and  H  the  E.  M.  F.  of 
the  standard  cell     Change  C  by  small  steps  and  plot  Q  and  d. 

FORM    OP   RECORD. 

Exercise  50.     To  determine  the  constant  of  a  ballistic  galvan- 
ometer. 

Date 

E         C         Q         d         c 


Galvanometer.  .  . 

Condenser  

Temperature 

Standard  cell.  . 

Mean  =:.... 

116.  Voltmeters  and  Ammeters.  Voltmeters  and  ammeters 
are  usually  portable  galvanometers  of  the  d'Arsonval  type.  Wall 
instruments  designed  to  measure  high  voltages  or  large  currents, 
as  in  electric  lighting  or  power  stations,  are  usually  attached  to 
the  switch  board  directly  and  give  continuous  indication  as  to 
the  pressure  and  strength  of  the  current  furnished.  Voltmeters 
and  ammeters  are  direct-reading  instruments,  that  is,  they  are  so 


Fig.  76. 


Fig.  77- 


calibrated  as  to  show  directly  upon  an  arbitrary  scale  the  differ- 
ence of  potential  existing,  or  the  current  flowing  between  any 
two  points  to  which  they  may  be  connected. 

The  best  known  instruments  of  this  class  are  those  designed 
by  Weston.  In  these  the  directing  couple  is  furnished  by  two  spiral 
springs  of  phosphor-bronze.  Rapid  damping  is  secured  by  the 
use  of  aluminum  frames  upon  which  the  coils  are  wound.  The 


I5O  PHYSICAL,    MEASUREMENTS 

voltmeter  (Fig.  76)  is  commonly  provided  with  two  scales,  one 
for  high  and  one  for  low  voltages.  In  the  figure  the  low  voltage 
terminal  on  the  negative  side  is  marked  15.  Back  of  the  needle 
is  a  mirror,  and  placing  the  eye  in  such  a  position  that  the  image 
of  the  needle  is  hidden  by  the  needle  itself  the  error,  due  to  par- 
allax, in  reading  the  scale  is  avoided.  The  Weston  instruments 
are  durable  and  remarkably  accurate  and  the  student  should  thor- 
oughly familiarize  himself  with  them  before  attempting  to  use 
them  independently. 

ELEMENTARY  EXERCISES. 

117.  Exercise  51.  Cells  in  Series  and  in  Parallel.  This 
simple  exercise,  designed  to  give  the  student  some  practice  in  the 
use  of  voltmeters  and  ammeters,  consists  in  the  study  of  the  ef- 
fect which  a  different  grouping  of  cells  has  upon  the  current  in 
an  electric  circuit.  Use  new  drv  cells  for  this  exercise. 


Fig.  78. 


(a)  Connect  a  voltmeter  V  to  the  terminals  of  a  cell  B,  being 
careful  to  join  the  binding  post  of  the  voltmeter,  marked  -)-,  to 
the  positive  pole  (Fig.  78).  Then  replace  the  cell  at  B,  first,  by 
two  cells  in  series  and,  second,  by  two  cells  in  parallel.  Take  the 
readings  in  the  three  different  cases  and  note  the  effect.  Should 
the  reading  with  the  two  cells  in  series  be  exactly  twice  that  ob- 
tained with  one  cell? 


EXERCISES 


(b)     Arrange  a  circuit,  consisting  of  a  cell  B,  a  rheostat  R 
of  about  TO  ohms  and  a  milammeter  A,  (Fig.  79).     Again  re- 


Fig.  79- 


place  the  cell  at  B,  first,  by  two  cells  in  series  and,  second,  by  two 
cells  in  parallel.  Note  the  effect  of  the  different  arrangements 
upon  the  reading  and  discuss  the  result. 


FORM    OF   RECORD. 


Exercise  51.     Study  the  effect  of  different  arrangements  of 
cells. 


Voltmeter  No, 
Ammeter  No. 


Voltmeter  reading 


Date 

Ammeter  reading 


One  cell 
Two  cells  in 

series 
Two  cells  in 

parallel 
Discussion  of  results. 


118.     Exercise  52.    Kirchhoff's  Laws.   Kirchhoff's  laws1  may 
be  studied  in  the  following  manner: 

(a)  Arrange  a  battery  B  in  series  with  three  rheostats  Rlt  R2 
and  Rs,  (Fig.  80).     Place  an  ammeter  or  milammeter  successively 
in  the  positions  a,  b,  c  and  d.     Take  the  readings  of  the  ammeter 
in  each  case.     Does  the  current  change  in  passing  through  a  re- 
sistance? 

(b)  Arrange  the  rheostats  in  parallel  (Fig.  81),  and  take  the 


1  College  Physics,  Article  271. 


152 


PHYSICAL    MEASUREMENTS 


readings  of  the  ammeter,  successively  in  positions  a,  b,  c  and  d. 
Discuss  the  results  as  examples  of  KirchhofFs  first  law : 


(c)     With  the  rheostats  in  series  (Fig.  80)  place  a  voltmeter 

t> 
b 


!i 


Fig.  So.  Fig.  81. 

successively  over  ab,  be,  cd  and  ad  and  read  the  differences  of 
potential  in  each  case. 

(d)  With  the  rheostats  in  parallel  (Fig.  81)  take  the  readings 
•of  the  voltmeter  over  ab,  ac,  and  ad.  Discuss  the  results  as  ex- 
amples of  KirchhofFs  second  law  : 


It  should  be  noted  that  an  ammeter  is  always  placed  in  series 
with  the  circuit  and  that  a  voltmeter  is  placed  as  a  shunt  between 
the  two  points  whose  potential  difference  is  to  be  measured. 


FORM    OF   RECORD. 

Exercise  52.     KirchhofFs  laws. 


Date. 


Position 
of  ammeter 

Rheostats 
in  series 

Position 
of  ammeter 

Rheostats 
in  parallel 

a 

a 

7 

d 

d 

Position 
of  voltmeter 

Position 
of  voltmeter 

ab 

ab 

be 

ac 

cd 

ad 

ad 

Discussion  of  results. 


ELEMENTARY    EXERCISES 


153 


ng.     Exercise  53.     Resistances  in  Series  and  in  Parallel. 

In  order  to  prove  the  laws  of  resistance1,  a  circuit  is  arranged 
consisting  of  a  battery  of  two  cells,  a  milammeter  and  three 
resistance  boxes  in  series,  with  resistances  R^  =  5,  R2  =  8  and 
R&  —  20  ohms  respectively.  Note  the  reading  of  the  ammeter  /. 
Take  two  boxes  out  of  the  circuit  and  adjust  the  resistance  R  in 
the  remaining  box  until  the  ammeter  reading  /  is  the  same  or 
nearly  the  same  as  before.  Be  careful  to  have  always  some  re- 
sistance in  the  circuit  so  as  to  avoid  shortcircuiting  the  cells. 
The  resistance  R  should  agree  with  the  equation 

R  =  Ri  -j-  Ra  -j-  Ra  ( l6o) 

Next,  arrange  the  three  resistance  boxes  in  parallel,  making 
the  resistance  of  each  15  ohms.  Note  the  reading  of  the  ammeter 
7V  Remove  one  of  the  boxes  and  make  the  resistances  of  the 
remaining  two  boxes  15  and  10  ohms  respectively.  Note  the  read- 
ing of  the  ammeter,  I2.  Finally  leave  only  one  box  in  the  circuit 
and  adjust  its  resistance  R'  until  the  current  I\  is  exactly  or  nearly 
equal  to  /± ;  adjust  again  to  R"  until  the  current  is  rz  =  h. 
The  results  should  agree  with  the  equation 

-4=2i-  (161) 


Give  a  sketch  of  the  arrangement  of  apparatus  for  each  of  the 
different  cases. 

FORM    OF   RECORD. 

Exercise  55.     To  study  the  laws  of  resistance. 

Date 

a.    Resistances  in  series : 


R  = 


I  — 

b.    Resistances  in  parallel : 


R'= 


Rs  = 


r,= 


1  College  Physics,  Article  276  and  277. 


154 


PHYSICAL    MEASUREMENTS 


MEASUREMENT  OF  RESISTANCE. 

120.  Exercise  54.  Resistance  by  Substitution.  The  ap- 
paratus is  arranged  as  shown  in  Fig.  82.  B  is  a  battery  of  con- 
stant E.  M.  F.,  x  the  unknown  re- 
sistance, R  a  resistance  box,  G  a 
galvanometer,  K  a  key  which  may 
connect  the  galvanometer  either  to 
x  or  to  R. 

The  battery  circuit  is  first  closed 
through  the  resistance  x,  and  the 
galvanometer.  If  the  deflection  be 
too  great,  the  controlling  magnet  or 
the  torsion  head  may  be  so  turned  as 
to  bring  the  deflection  back  upon 
the  scale,  or  a  shunt  may  be  applied 
to  the  galvanometer.  Do  not  use  a 


Fig.  82. 


high  resistance  in  the  circuit  instead  of  a  shunt    (Why?). 

Assuming  that  the  deflections  are  porportional  to  the  current  we 
have 


i  =  c  1 1  =  c  . 


(162) 


where  r  is  the  resistance  of  the  whole  circuit  except  x,  and  c  is  the 
proportionality  factor.  Next  send  the  current  through  R  instead 
of  .r;  then 


E 


R  +  r 


(163) 


If  now  R  be  adjusted  until  the  deflections  in  the  two  cases  are 
equal,  then 

x  =  R.  (164) 


If  R  cannot  be  adjusted  so  as  to  produce  exactly  the  same  de- 
flection as  x,  interpolate  between  the  two  nearest  values  above  and 
below.  Measure  three  resistances  separately  and  in  series. 


RESISTANCE 


155 


FORM    OF  RECORD. 

Exercise  54.     To  measure  three  resistances  by  substitution. 

Date 

I     Deflection  with  R2 

•   •    •   • 4 

,  ohms. 


Galvanometer 

Resistance  of 

Deflection  with  x 


Temperature 
Resistance  box. 
Deflection  with 


+  Xi  -f-  X*  = 


121.  Exercise  55.  Resistance  by  Voltmeter  and  Ammeter. 
This  exercise  consists  in  measuring  at  the  same  time,  the  cur- 
rent flowing  through  a  wire  and  the  difference  of  potential 
at  its  terminal  points.  Then  if  x  be  the  resistance  of  the  wire, 
we  have  by  Ohm's  law 

V 


x-=. 


(165) 


The  potential  difference  at  the  terminals  is  measured  by  means 
of  a  voltmeter  whose  resis- 
tance must  be  very  high  in 
comparison  with  the  resist- 
ance to  be  measured,  as  other- 
wise the  current  through  the 

resistance  x,  will  not  be  the  * 

total   current   I,  measured  by 


<£> 


b  x  a 

Fig.  83. 
the  ammeter.    This  is  easily  seen  by  applying  the  shunt  rule, 


(166) 


where  7?v  is  the  resistance  of  the  voltmeter.  If  the  resistance  x 
is  too  large,  it  is  better  to  include  the  ameter,  whose  resistance 
is  very  small,  together  with  the  resistance  x  between  the  terminals 
of  the  voltmeter,  as  shown  by  the  dotted  lines  in  Fig.  83,  and  then 
subtract  the  resistance  of  the  ammeter  from  the  resulting  value 
of  x. 

Measure  the  resistance  of  an  incandescent  lamp.  Put  a  rheostat 
in  series  with  the  lamp  and  the  ammeter.  Cut  out  resistance 
from  the  rheostat  step  by  step  until  the  lamp  has  reached  its  full 
candle  power.  Observe  at  each  step  the  readings  of  the  volt- 
meter and  ammeter.  Calculate  the  resistance  for  each  current. 


156 


PHYSICAL    MEASUREMENTS 


and  the  watts  absorbed  by  the  lamp;  also  the  watts  per  candle 
when  the  lamp  is  burning  at  its  normal  voltage. 

FORM    OF   RECORD. 

Exercise  55.       To  measure  the  resistance  of  an  incandescent 
lamp  by  voltmeter  and  ammeter. 

Date    

Voltmeter  No Milammeter 

Name  of  Lamp '    Candle   power    

V  1  R  Watts 


Watts  per  candle  = 

122.  Exercise  56.  Very  High  Resistance  by  Direct  De- 
flection. This  method  is  a  modification  of  the  last.  Instead  of  an 
ammeter,  a  galvanometer  (Fig.  84)  is  used  whose  figure  of  merit 
is  determined  as  in  Exercise  49.  The  formula  for  the  resistance 
then  becomes 

x=j-d—g,  (167) 

since  /  —  /  d. 

Usually  g,  the  resistance  of  the  galvanometer,  is  negligible  in 
comparison  with  the  high  resistance  X.  High  resistances  of  this 

kind  are  insulation  resistances,  as 
for  example,  the  resistance  offered 
to  the  passage  of  a  current  by  the 
insulation  of  a  cable  or  of  a  con- 
denser. It  is  advisable  to  insert  a 
high  resistance  H.R.  in  series  with 
the  galvanometer,  in  order  to  pro- 
tect the  instrument  from  excessive 
currents  in  case  of  a  break  in  the 
insulation.  If  the  deflections  vary 
notably  with  the  time,  it  is  advis- 
able to  keep  the  key  K  closed  and 
g4  take  a  series  of  readings  at  definite 

time  intervals.  //  X  represents  the  resistance  of  a  condenser, 
short  circuit  the  galvanometer  by  means  of  a  shunt  key,  before 
closing  the  key  K,  and  afterwards  open  the  short  circuiting  key 
to  observe  the  steady  deflection. 


RESISTANCE 


157 


Date. 

.... 

Time 

V 

d 

X 

FORM    OF   RECORD. 

Exercise  $6.  To  measure  the  resistance  offered  by  the  insula- 
tion of  a  commercial  condenser.  ' 

Condenser 

Galvanometer 

Figure  of  merit,  (Exercise  49) 

Resistance  of  the  galvanometer 

123.  The  Wheatstone  Bridge.  The  most  accurate  methods 
for  measuring  resistance  are  based  upon  the  Wheatstone  bridge 
principle.  In  its  simplest  form  the  Wheatstone  bridge  consists  of 
a  network  of  six  conductors  joining  four  points,  A,  B,  C,  D, 
(Fig.  85),  so  arranged  that  each  point  is  joined  to  each  of  the 
other  three  points  by  separate  conductors. 

Let  one  of  the  conductors  contain  a  source  of  E.  M.  F. ;  four 
of  the  others  will  form  a  divided 
circuit  while  the  remaining  one, 
containing  a  galvanometer,  will 
form  a  bridge  between  the  two' 
parallel  conductors.  Let  R^, 
R2,  Rs,  R±  be  the  resistances 
forming  the  four  branches  of  the 
divided  circuit,  and  suppose  them 
to  be  so  adjusted  that  no  cur- 
rent flows  through  the  galvan- 
ometer—  Zero  method.  Then 
it  may  be  shown  that  the  resist- 
ances satisfy  the  relation 

R^ 


E*. 
R* 


(168) 


For  let  the  potentials  of  the  four  points  be  represented  by 
F&,  Fb,  Fc,  Fd,  then  since  there  is  the  same  fall  of  potential  be- 
tween A  and  B,  whether  we  pass  by  one  route  or  the  other,  and 
since  with  a  constant  current  the  fall  of  potential  is  at  all  times 
proportional  to  the  resistance  passed  over,  we  have 


Fa—  Fc 


and 


Fa—  Fj 
Fa  —  V  b 


R, 


(169) 


158  PHYSICAL    MEASUREMENTS 

But  since  no  current  passes  through  the  galvanometer 

P.—  Fc=Fa—  Fd 

therefore 

R 


whence 

^=#r  (I7I) 

From  the  above  relation  it  is  evident  that  if  three  of  the  resist- 
ances be  known  the  fourth  may  at  once  be  determined.  In  fact  it 
is  necessary  to  know  but  one  resistance  and  the  ratio  between  the 
other  two.  Since  the  current  through  the  parallel  branches  should 
become  steady  before  the  potential  at  C  and  D  are  tested,  it  is 
necessary  to  close  the  galvanometer  key  last  in  every  case. 
A  successive  contact  key  is  best  adapted  to  this  work.  What 
would  be  the  effect  of  selfinductance  in  one  of  the  branches? 

It  will  be  found  advantageous  to  follow  Maxwell's  rule  :x  "Of 
the  two  resistances — that  of  the  battery  and  that  of  the  galva- 
nometer— connect  the  greater  resistance  so  as  to  join  the  two 
greatest  to  the  two  least  of  the  four  other  resistances."  This 
insures  the  greatest  sensitiveness  of  the  apparatus. 

vSince  both  the  potentials  at  C  and  at  D  depend  directly  upon 
the  difference  of  potential  between  A  and  B,  the  absolute  value  of 
the  latter  or  a  slow  change  in  the  E.  M.  F.  of  the  battery  will  not 
affect  the  balance  of  the  galvanometer.  A  Leclanche  cell  may 
therefore  be  used  as  the  source  of  current. 

The  temperature  of  the  room  must  be  carefully  noted. 

124.  Exercise  57.  Resistance  by  Wheatstone  Bridge  Box. 
A  very  convenient  form  of  apparatus  for  the  measurement  of 
resistance  by  the  Wheatstone  bridge  method  is  a  box  (Fig.  86), 
containing  three  known  resistances  connected  in  series,  one  of 
which  may  be  given  any  value  between  one  ohm  and  the  maximum, 


•Maxwell,  Electricity  and  Magnetism,  jd  edition,  Vol.  I,  p.  478. 


RESISTANCE 


159 


00000000 

.50    20    10    10     5      2      1       1 


usually  10,000  ohms. 
The  other  two  resist- 
ances form  the  pro- 
portional arms  of  the 
Wheatstone  bridge 
and  contain  but  a  few 
coils,  usually  I,  10, 
100  and  1000  ohms. 
In  this  way  the  ratio 
between  the  known 
and  the  unknown  re- 
sistances  may  be  Fig.  86. 

varied  from  1000  to  o.ooi.  The  galvanometer  is  generally  con- 
nected to  the  outer  ends  of  the  proportional  coils  and  the  battery 
to  the  remaining  two  binding  posts.  Frequently  the  two  keys  and 
a  galvanometer  are  included  in  the  case,  thus  making  it  a  very 
compact  and  convenient  instrument  for  the  measurement  of  re- 
sistance. (Fig.  87.)  This  form  of  Wheatstone  bridge  is  also 
known  as  the  Postoffice  Box. 

In  practice,  especially 
where  the  unknown  resist- 
ance is  not  even  approxi- 
mately known,  it  is  well  to 
begin  with  equal  resistances 
in  the  proportional  arms  R2 
and  R±.  If  the  unknown  re- 
sistance RI  is  too  large  the 
deflection  on  closing  the 
galvanometer  will  be  in  one 
direction,  if  too  small  it 
will  be  in  the  opposite  di- 
rection. First  find  two 
values  for  R3  which  change 
the  direction  of  the  de- 
flection of  the  galvanom- 
eter, and  keep  in  mind  the  meaning  of  the  direction  of  the 
deflection.  After  having  thus  found  the  approximate  value 
of  the  unknown  resistance,  change  the  ratio  of  the  propor- 
tional arms  so  as  to  obtain  the  smallest  value  of  the  ratio  R2/R4 


Fig.  87. 


i6o 


PHYSICAL    MEASUREMENTS 


that  R3  will  allow  and  still  produce  a  balance,  and  then  make  the 
final  determination.  Sometimes  it  may  be  necessary  to  interpolate 
between  the  values  of  Rs.  Measure  the  resistance  of  three  pieces 
of  wire,  and  calculate  the  resistivity  of  each  metal  from  the  form- 
ula 

P  =  R  j~  ,  (172) 

where  R  is  the  resistance,  /  the  length  and  a  the  cross-sectional 
area  of  the  wire. 

FORM    OF  RECORD. 

Exercise  57.     To  determine  the  rcsisthnty  of  three  metals. 


Postoffice  box  No 
Galvanometer 

R3 

Date 

Temp. 
R=RA     I 

of  w 
diam. 

ire 

Temp    of    room  

Specimen  of  wire 

R3 

R, 

a 

P 

::::::;::::: 

125.  Exercise  58.  Resistance  by  Slide- Wire  Bridge.  From 
equation  (171),  it  is  seen  that  only  the  ratio  R3/R±,  and  one  re- 
sistance R2,  need  be  known,  to  effect  the  measurement  of  an  un- 
known resistance.  The  ratio  may  be  furnished  by  the  two  parts  of 
a  wire  of  uniform  cross-section.  In  the  slide-wire  bridge  the  sum 
of  the  lengths  of  the  two  wires  representing  Rs  and  R±  is  kept  con- 
stant, usually  i ocx)  mms,  and  the  ratio  of  their  lengths  is  changed 
by  moving  one  of  the  galvanometer  terminals  along  the  wire.  For 

this  purpose  a  contact 
maker  Kf  (Fig.  88). 
is  substituted  for  the 
galvanometer  key  K2 
in  the  Wheatstone 
bridge.  If  on  closing 
K  and  making  contact 
at  Kr ,  no  current 
flows  through  the  gal- 
vanometer, we  have, 
neglecting  the  very 
low  resistance  of  the 
copper  straps, 

£-  <'"> 


Fig.  88. 


RESISTANCE  l6l 

or  letting  the  reading  at  D  on  the  wire  equal  o^  mms,  then 


(I74) 


To  obtain  accurate  results  we  must  interchange  x  and  R.  in 
order  to  correct  for  possible  errors  due  to  variations  in  the  cross- 
section  of  the  wire,  or  to  unsymmetrical  placing  of  the  scale, 
or  to  a  constant  error  in  the  reading  of  the  position  of  the  contact 
maker.  After  exchanging  x  and  R,  a  new  reading,  a2,  is  obtained  ; 
then 


or  combining  (174)  and  (175) 


TOCO -+-(at —  «•>)  ,     -. 

(176) 


R       1000 —  (<?! —  a2) 

As  shown  on  page  8,  the  error-s  of  observation  influence  the 
result  least  when  the  contact  maker  is  at  the  middle  of  the  wire. 
That  is  when  x  and  R  are  equal. 

In  some  bridges  four  gaps  are  provided  in  the  large  copper 
strips  for  the  insertion  of  resistance.  In  such  bridges  the  two 
inner  gaps  are  for  the  resistances  of  x  and  R,  while  the  outer 
ones  may  be  used  to  insert  two  nearly  equal  resistance  coils  which 
serve  as  extensions  of  the  slide  wire,  thus  enabling  the  experi- 
menter to  determine  the  ratio  of  the  two  lengths  with  greater 
accuracy.  These  additional  coils  must  be  determined  in  terms 
of  mms  of  the  bridge  wire.1 

Avoid  moving  the  contact  maker  K'  while  pressed  down. 

A  slight  pressure  should  suffice  to  make  contact.    If  not,  the  key 
and  wire  should  be  cleaned. 


1  Carhart  and  Patterson,  Electrical  Measurements,  pp.  58-64. 


162 


PHYSICAL    MEASUREMENTS 


FORM    OF 'RECORD. 

Exercise  58.     To  measure  the  resistance  of  an  electromagnet. 
Study  effect  of  self-inductance  by  closing  K'  before  K. 


Slide-wire  bridge  No Date. 

Galvanometer Temperature 

R  ai  a* 


126.  Exercise  59.  Resistance  of  a  Galvanometer.  —  Thom- 
son's Method.  The  principle  of  the  Wheatstone  bridge  may  be 
applied  to  the  measurement  of  the  resistance  of  a  galvanometer. 
Here  the  galvanometer  is  put  in  one  of  the  four  arms  of  the  bridge 
and  the  battery  circuit  closed,  producing  a  deflection  of  the  gal- 
vanometer. If  the  deflection  is  too  great,  a  resistance  may  be  in- 
troduced into  the  battery  circuit,  to  reduce  the  current  —  or  the 
controlling  magnet  may  be  lowered  or  the  torsion  head  twisted 
until  the  deflection  comes  once  more  upon  the  scale.  Instead  of 
a  galvanometer  as  in  the  ordinary  arrangement,  a  key  is  inserted, 
shown  as  K'  in  Fig.  89,  or  the  contact  maker  may  be  used  as 
before.  If  with  a  steady  deffection  of  the  galvanometer  this 
key  be  closed  and  a  current  pass  between  C  and  D  in  either 
direction  then  the  deflection  of  the  galvanometer  will  change. 
Only  when  there  is  no  current  through  K'  ,  will  the  scale  read- 

ing remain  constant. 
When  a  constant 
scale  reading  with 
K'  open  or  closed  is 
B  attained,  the  condi- 
tion for  a  balance  in 
the  bridge  is  fulfilled. 


Fig.  89. 


looo  — a 


ohms. 


(177) 


In  this  case  also  the  best  results  are  obtained  by  making  R  =  g. 
Exchange  g  and  R  and  apply  equation  (176). 

At  first  any  resistance  Rlt  may  be  taken  and  the  balance  sought, 


RESISTANCE 


I63 


simply  to  find  g  approximately.  Then  make  R  nearly  equal  to  g 
and  repeat  with  greater  care.  Make  three  separate  determinations 
by  varying  R.  Must  a  cell  of  constant  E.  M.  F.  be  used  in  this 
experiment  ? 

FORM   OF  RECORD. 

Exercise  59.     To  measure  the  resistance  of  a  galvanometer. 

Date 

Temperature Galvanometer  No 

Preliminary  Final 


*,=  . 

a  =  . 
9  =  • 


R 

tfi 

a, 

g 

127.  Exercise  60.  Resistance  of  a  Galvanometer. — Second 
Method.  The  resistance  of  a  galvanometer  may  also  be  deter- 
mined by  the  application  of  the  principle  involved  in  Kxercise  48. 
The  apparatus  is  arranged  as  shown  in  Fig.  71.  With  no  resist- 
ance in  R2  the  deflection  of  the  galvanometer  is  observed  for  a 
length  of  bridge  wire  o^,  of  about  30  cms.  The  length  of  the  wire 
is  next  made  about  twice  as  large,  that  is  60  cms  approximately, 
and  sufficient  resistance  is  added  in  R2  to  make  the  deflection  of 
the  galvanometer  approximately  equal  to  the  first  deflection.  Fi- 
nally move  the  contact  key  along  the  wire  until  the  deflections  are 
exactly  equal.  Call  this  reading  of  the  bridge  a2.  Then  neg- 
lecting the  resistance  of  the  wire  in  comparison  with  that  of  the 
galvanometer 


9  = 


(179) 


Make  three  separate  determinations  of  g. 


i64 


PHYSICAL    MEASUREMENTS 
FORM    OF   RECORD. 


.  Exercise  60.     To  measure  the  resistance  of  a  galvanometer  by 
second  method. 


Galvanometer  No , 


Date 
Temperature 


9  — 


128.  Exercise  61.  Resistance  of  ah' Electrolyte.  When  a 
constant  current  is  sent  through  an  electrolyte,  polarization  pro- 
duces a  counter  E.  M.  F.  and  so  renders  the  principle  of  Wheat- 
stone's  bridge  inapplicable.  Polarization  may  be  avoided  by  the 
use  of  rapidly  alternating  currents.  The  simplest  source  of 
such  alternating  currents  is  a  small  induction  coil  of  high  fre- 
quency. The  galvanometer 
must  be  replaced  by  an  in- 
strument capable  of  detect- 
ing alternating  currents.  A 
telephone  receiver  of  from 
10  to  30  ohms  resistance  is 
best  adapted  for  this  pur- 
pose. Otherwise  the  man- 
ipulation is  similar  to  that 
described  in  article  125. 

^^P^      -    m        Vary  the  resistance  or  move 

^^  Jr          the  contact  maker  until  the 


Fig.  90. 


sound  in  the  telephone 
ceases  or  becomes  a  mini- 
mum. 

The  most  convenient  form  of  bridge  is  one  which  has  the  wire 
wound  on  an  insulating  cylinder  and  in  which  contact  is  made  by 
a  small  grooved  wheel  rolling  upon  the  wire  as  the  cylinder  is 
revolved.  This  form  of  bridge,  (Fig.  90),  is  due  to  Kohlrausch. 
It  is  evident  that  self-induction  must  be  avoided  as  far  as 
possible  in  all  parts  of  the  apparatus.  The  electrolyte  is  con- 


RESISTANCE  165 

tained  in  a  glass  vessel  furnished  with  two  large  platinum  elec- 
trodes covered  with  platinum  black.     If  the  vessel  has  a  cylin- 
drical or  prismatic  form  (Fig.  91)  the  resistivity  may  be 
readily  determined  from  the  formula  given  in  Exercise         f 
57.     In  electrolytes,  however,  the  result  is  usually  ex-        ,4, 
pressed  in  terms  of  the  conductivity 


*-  L  =  4- 

"  P        a#  • 

If  the  form  of  the  vessel  be 
irregular  (Fig.  92),  comparison 
must  be  made  with  an  electrolyte 
of  known  conductivity  k.  As 
such  a  standard  we  may  take  a 
saturated  solution  of  NaCl,  sp.  gr.  1,201  at 
18°  C.  For  such  a  solution 

k  =  0.216  [i  +  0.023  (t-i8)  ]  ohm- 1  cm-*. 

See  also  Table  XV  for  other  solutions  which  may  be  used  as 
standards.    Letting  the  resistance  of  a  standard  NaCl  solution  be 

R  and  that  of  an  electrolyte  RK,  then  -±L__£L   and  kK  the  conduc- 
tivity sought  is  given  by  the  relation 


(180) 


The  quantity  Rk  is  a  constant  for  each  vessel  and  is  called  the 
resistance  capacity  of  the  vessel. 

Since  the  temperature  coefficient  of  electrolytes  is  very  large, 
great  care  should  be  taken  in  the  determination  of  the  temperature. 

Frequently,  especially  in  German  tables,  the  conductivity  is 
referred  to  that  of  mercury  at  o°C.  Since  the  resistivity  of  mer- 
cury at  o°C  is  1/10630  ohm  cm,  its  conductivity  is  10630  ohm'1 
cnr1.  To  find  k  in  the  terms  of  mercury  at  o°C  divide  the  value 
found  from  (180)  by  10630.  The  conductivity  of  electrolytes 
is  usually  expressed  either  as  molecular  conductivity  or  as  equival- 
ent conductivity. 


i66 


PHYSICAL    MEASUREMENTS 


The  "molecular  conductivity"  ^  is  defined  by  the  equation 
p  =  k/m  where  m  is  the  number  of  gram  molecules1  of  the  sub- 
stance dissolved  in  one  cubic  centimeter  of  the  solution. 

The  molecular  concentration  is,  however,  generally  expressed 
as  the  number  N  of  gram  molecules,  dissolved  in  a  liter  of  the 
solution,  so  that  N  equals  1000  m.  From  this  follows 


1000 


(181) 


looo/N  is  the  number  of  cubic  centimeters  containing  one  gram 
molecule  of  the  dissolved  substance.  The  conductance  of  a  cubic 
centimeter  of  a  body  between  two  opposite  faces  of  the  cube  is 
numerically  equal  to  the  conductivity  of  the  substance  (equation 
172).  If,  therefore,  that  quantity  of  a  solution  which  contains  one 
gram  molecule  of  the  dissolved  substance  be  brought  between  two 
electrodes,  one  centimeter  apart  and  forming  the  sides  of  the 
containing  vessel,  the  conductance  observed  will  be  numerically 
equal  to  the  molecular  conductivity  of  the  electrolyte. 

The  equivalent  conductivity  A,  is  found  from  the  molecular 
conductivity  by  dividing  the  latter  by  the  valence  of  the  ions, 
t.  e.,  in  case  of  acids  by  their  basicidity,  in  bases  by  their  acidity, 
in  salts  by  the  number  of  acid  or  basic  valences  present.  Equiva- 
lent conductivity  is  a  quantity  of  great  importance  in  the  modern 
electrolytic  theory.  V 

FORM   OF   RECORD. 

Exercise  61.  To  determine  the  specific  and  molecular  conduc- 
tivities of  a  solution  of  a  salt,  using  three  different  dilutions. 

Date.  . 


t° 


Standard  : 

I 

di 

0» 

r2 

R 

Solution 

ai 

Wl»  ^-^ 

m  = 

kR= 

m  = 

Final  measurements 


1  A  gram  molecule  of  a  substance  is  a  mass  of  the  substance,  in  grams, 
equal  to  the  molecular  weight  of  the  substance. 


POTENTIAL  DIFFERENCE  167 

ELECTROMOTIVE   FORCE  AND   POTENTIAL  DIFFERENCE. 

129.  The  Voltmeter.     The  simplest  method  for  measuring  a 
difference  of  potential  between  two  points  is  to  connect  them  to  a 
voltmeter,  or  to  any  instrument  whose  graduation  allows  us  to 
read  off  the  potential  difference  in  volts  at  its  terminals.  The  volt- 
meter was  employed  for  this  purpose  in  Exercises  51,  52  and  55. 
The  student  should  bear  in  mind  that  the  voltmeter  must  at  all 
times  possess  a  very  high  resistance  Rv,  if  its  indications  are  to  be 
reliable.    Thus  in  measuring  the  drop  in  potential  over  a  known 
resistance   R,   the   instrument   actually   gives   not   E  =  IR,   but 

£'  — /  t  R  -  Rv  .      Now  it  is  clear  that  E'  can  equal  E  only  when 
*  R  +  Rr 

Rv  is  so  large  that  R  may  be  neglected  in  comparison. 

130.  Exercise    62.     Electromotive   Force   of  a    Cell.     The 
apparatus  (Fig.  93),  consists  of  a  battery  B,  of  E.  M.  F.  higher 
than  that  of  the  cell  to  be  measured,  a 

resistance  R^  and  a  voltmeter.  The 
cell  B',  whose  E.  M.  F.  is  to  be  deter- 
mined, is  joined  in  parallel  with  the 
voltmeter,  so  that  the  E.  M.  F.  is  in 
the  same  sense  as  that  of  the  cell  B. 
Close  key  K'  and  adjust  the  resistance 
RI,  until  on  closing  key  K,  the  reading 
of  the  voltmeter  does  not  change.  The 
current  from  B  produces  a  difference  Fig.  93. 

of  potential  IRV  at  the  terminals  of  the  voltmeter,  where  Rv  is  the 
resistance  of  the  voltmeter.  If  this  P.  D.  be  smaller  than  the  E. 
M.  F.  of  the  cell  B' ',  then  upon  closing  K  an  additional  current, 
taken  from  this  cell,  will  flow  -through  the  voltmeter  and  increase 
the  reading.  If  IRV  be  larger  than  the  E.  M.  F.  of  B'  a  part  of 
the  current  I  will  be  shunted  through  Bf  and  the  reading  of  the 
voltmeter  be  decreased  upon  closing  K.  Only  if  IRV  equals  the 
E.  M.  F.  of  B'  will  no  current  flow  through  the  cell  upon  closing 
K;  and  the  voltmeter  reading  which  is  IRV  measures  then  directly 
the  E.  M.  F.  of  B'  no  matter  how  large  the  resistance  of  the  volt- 
meter is. 


i68 


PHYSICAL    MEASUREMENTS 


To  increase  the  accuracy  of  the  method  a  galvanometer  more 
sensitive  than  the  voltmeter  may  be  inserted  in  series  with.  Bf  and 
R!  adjusted  until  the  galvanometer  shows  no  deflection  on  closing 
K.  A  high  resistance  in  series  with  B' ' ,  may  be  used  at  first  to 
prevent  the  passage  of  too  large  a  current  through  the  cell,  but  it 
should  be  removed  before  making  the  final  adjustment. 

By  placing  one  or  more  standard  cells  at  B'  this  method  is  ad- 
mirably suited  for  the  calibration  of  voltmeters  at  points  corre- 
sponding to  the  electromotive  forces  inserted. 


FORM   OF   RECORD. 

Exercise  62.     To  measure  the  E.  M.  F.  of  five  different  cells, 
first  separately  and  afterward  all  joined  in  series. 


Voltmeter  No, 
Name  of  cell. 


Date 

Voltmeter  reading, 


131.     Exercise  63.     Electromotive  Force  by  Potentiometer 

Method.    In  measurements  of  E.  M.  F.  or  of  potential  difference 

where  a  high  degree  of  accuracy  is  required,  it  is  necessary  to 

compare  the  E.  M.  F.  to  be  measured  with  that  of  a  standard  cell. 

The  arrangement  of  the  apparatus  is  shown  in  Fig.  94.     The 

principle  of  the  method  consists 
in  producing  by  means  of  a  cur- 
rent from  the  battery  B,  a  poten- 
tial difference  at  the  terminals 
of  a  resistance  R,  equal  to  the  E. 
M.  F.  of  the  cell  B'  whose  elec- 
tromotive force  £  is  to  be  meas- 
ured. The  two  electromotive 
forces  should  be  so  arranged  as 
to  oppose  each  other  in  the  gal- 
vanometer branch.  If  now  the  keys  K  and  Kf  be  closed,  no  cur- 


B 


H.R 

It 

R 

rK 

*ll 

K 

94- 


POTENTIAL  DIFFERENCE  169- 

rent  will  flow  through  the  galvanometer  when  £  equals  i  R.  In 
practice  a  standard  cell  is  first  placed  at  B' '.  Let  its  known  E.  M. 
F.  be  £A ;  then  after  the  resistance  R  has  been  so  adjusted  that  no 
current  flows  through  the  galvanometer  on  closing  the  keys,  we 
have 

El  =  'hR1  (182) 

The  cell  under  experiment  is  next  set  in  at  B'  and  after  a  new 
adjustment  of  resistance  for  no  current,  we  have 

E*=hRs  (183) 

and  under  the  condition  that  f±  =  iz,  we  have  at  once 

£,  =  &.   ^  •  (184) 

The  condition  i±  =  i2  is  fulfilled  when  the  battery  B  has  a  con- 
stant E.  M.  F.,  and  when  the  sum  of  the  resistances  R  and  R' 
remains  constant  throughout  the  experiment.  This  sum  should 
be  at  least  2000  ohms.  It  is  moreover  evident  that  the  E.  M.  F. 
of  B  must  be  greater  than  that  of  B' '.  A  pair  of  Daniell  cells  or 
a  storage  battery  is  best  for  this  purpose.  In  this  latter  case  the 
auxiliary  circuit  should  be  closed  for  about  half  an  hour  before 
beginning  the  measurements  and  be  kept  closed  throughout  the 
experiment.  If  the  E.  M.  F.  of  ordinary  open  circuit  cells  is  to 
be  measured,  a  freshly  charged  Leclanche  element  will  do  very 
well  as  the  high  resistance  of  R  +  R'  prevents  it  from  polarizing 
appreciably,  if  the  key  K  is  kept  closed  for  but  an  instant.  The 
high  resistance  H.R.  is  inserted  in  the  galvanometer  circuit  to 
prevent  the  passage  of  large  currents  through  the  instrument.  It 
should  be  cut  out  when  a  balance  is  nearly  obtained. 

It  is  most  convenient  to  use  two  boxes  of  the  same  kind  in  R 
and  R'  and  keep  the  sum  R  -\-R'  equal  to  the  sum  of  the  resist- 
ances in  one  box.  In  this  case  it  is  easy  to  check  the  constancy  of 
the  sum,  since,  whenever  a  plug  is  removed  in  one  box  it  has  to 
be  inserted  in  the  corresponding  hole  of  the  other. 


PHYSICAL    MEASUREMENTS 


Name  of  cell 

R 

R' 

E 

xes 

No 

:andard  cell. 

It  is  clear  that  a  wire  of  constant  length  and  uniform  crosssec- 
tion  may  be  substituted  for  the  two  resistances  R  and  R' ,  and  a 
balance  obtained  by  shifting  one  terminal  of  the  galvanometer 
along  the  wire.  If  the  readings  are  %  and  a2  in  the  two  cases, 
then 

&=a.*-  ess) 

In  this  case  however,  care  must  be  taken  to  avoid  heating  the 
wire,  and  to  this  end  the  key  K  should  be  kept  closed  but  an  in- 
stant at  a  time. 

FORM   OF  RECORD. 

Exercise  63.    To  measure  the  E.  M.  F.  of  five  cells,  separately 

and  in  series. 

Date, 

Name  of  cell 
Galvanometer 
Resistance    boxes 
Standard  cell  No 
Temperature 
E.  M.  F.  of 

.  132.  The  Potentiometer.  The  frequent  application  of  the 
principle  of  the  potentiometer  to  the  measurement  of  the  elec- 
tromotive forces  of  cells,  of  thermo-couples,  or  of  the  potential 
difference  at  the  terminals  of  a  standard  resistance  due  to  the 
passage  of  a  current,  has  led  to  the  construction  of  resistance 
boxes  of  special  design  called  potentiometers.  The  principle  is 
the  same  as  that  given  above  and  a  description  of  one  of  the  many 
forms  on  the  market  will  explain  the  method  of  using  such  an  in- 
strument. 

A  standard  cell  is  connected  through  two  binding  posts  N  (Fig. 

95)  to  the  dial  switches, 
which  allow  the  galvan- 
ometer circuit  to  be  con- 
nected to  a  definite  part 
of  the  total  constant  re- 
sistance of  the  potenti- 
ometer. The  switches 
must  be  so  arranged  that 

Fig.  95.  this   portion   of   the   re- 

sistance in  the  galvanometer  circuit  reads  to  some  decimal  mul- 


POTENTIAL  DIFFERENCE; 


171 


tiple  of  the  electromotive  force  of  the  standard  cell.  The  aux- 
liary  current,  entering  the  instrument  at  B,  is  then  adjusted  by 
means  of  an  external  resistance,  until  a  balance  of  the  galvanom- 
eter is  obtained.  By  the  turning  of  the  switch  in  the  upper  left 
hand  corner,  the  unknown  potential  difference  at  X  is  substituted 
for  the  standard  cell  and  the  dial  switches  again  adjusted  until  a 
balance  is  once  more  attained.  The  potential  difference  can  then 
be  read  off  directly  from  the  resistance  indicated  by  the  dials. 
In  some  instruments  the  standard  cell  is  connected  to  a  separate 


AT..- 


Fig.  96. 

resistance  outside  the  potentiometer  proper,  so  that  at  any  time 
the  balance  with  the  standard  cell  may  be  checked  without  dis- 
turbing the  position  of  the  dials. 

133.  Exercise  64.  Calibration  of  a  Voltmeter.  This  method 
also  consists  in  balancing  the  E.  M.  F.  of  one  or  more  standard 
cells  against  the  potential  difference  at  the  terminals  of  a  resist- 
ance traversed  by  a  current.  Let  V  be  the  difference  of  potential 
at  the  terminals  of  the  voltmeter,  £  be  the  E.  M.  F.  of  the  stand- 
ard cells,  and  i  the  current  flowing  through  R  and  R' ;  then 


")  and  E  =  iR, 


or 


(186) 
(187) 


PHYSICAL    MEASUREMENTS 


The  arrangement  is  that  shown 
in  Fig.  97.  The  resistance  r  is  in- 
troduced to  vary  the  readings  of 
the  voltmeter,  but  it  should  be  noted 
that  the  voltmeter  reading  will  then 
vary  more  or  less  with  the  change 
of  R  or  R'.  If  the  accuracy  of  any 
individual  reading  of  the  instru- 
ment is  to  be  tested  a  repeated  ad- 
justment of  r  may  be  necessary.  In 
this  case  it  would  be  more  conven- 
ient to  adjust  the  number  of  cells 
in  B  so  as  to  give,  as  nearly  as  pos- 
Fi£-  97-  sible,  the  desired  reading  or  to  keep 

the  sum  R  plus  R'  constant,  which  in  general  is  not  necessary. 

The  voltmeter  must  be  kept  in  the  circuit  all  the  time  during 
the  experiment.  A  high  resistance  in  the  galvanometer  branch 
before  the  final  balance  is  reached  is  also  advisable  here.  The  volt- 
meter correction  is  the  amount  to  be  added  to  the  voltmeter  read- 
ing to  make  it  correspond  to  the  computed  value  of  V .  For  the 
calibration  of  the  low  readings  of  a  voltmeter  see  also  Exercise 
62.  Plot  voltmeter  readings  and  corrections. 

FORM    OF   RECORD. 


D 

ate.  .  .  . 

Galvanometer 

Reading  of 
voltmeter 

R 

R' 

V 

Correction 

Standard  cell  No 

Resistance    boxes 

Temnerature     . 

134.  Exercise  65.  Thermoelectromotive  Force  of  a  Ther- 
moelement. A  thermoelement1  is  usually  used  in  such  a  way 
as  to  keep  one  of  the  junctions  at  a  constant  temperature,  pre- 
ferably o°C.,  while  the  other  junction  is  brought  into  contact  with 
the  point  whose  temperature  is  to  be  determined.  If  immersed  in 
a  liquid  the  junctions,  with  the  wires  well  insulated  from  each 


1  College  Physics,  Article  303. 


POTENTIAL 


173 


other,  should  be  surrounded  by  thin  glass  or  porcelain  tubes  in 
order  to  avoid  disturbances  due  to  voltaic  action.  For  better 
conduction  of  heat  the  tubes  may  be  rilled  with  pure  paraffin  oil  or 
kerosene. 

Insert  one  of  the  terminal  tubes  in  ice  water  and  place  the  other 
in  a  water  bath  to  be  heated.  The  tempera- 
tures should  be  determined  by  means  of  ac- 
curately calibrated  thermometers.  Con- 
nect the  free  terminals  of  the  couple  to  a 
low  resistance  galvanometer,  whose  volt 
sensitiveness  is  known.  Vary  the  tempera- 
ture of  the  water  bath  by  eight  or  ten  steps, 
between  o°  and  ioo°C.,  and  plot  a  curve, 
using  temperatures  as  abscissae  and  elec- 
tromotive forces  as  ordinates.  It  is  us- 
ually most  convenient  to  use  the  critical  re- 
sistance for  the  galvanometer  circuit,  but 

in  case  it  is  found  necessary  to  increase  the  resistance  of  the  cir- 
cuit, then  the  volt  sensitiveness  of  the  instrument  must  be  deter- 
mined for  the  particular  resistance  of  the  circuit  as  used.  For 
more  accurate  work  a  potentiometer,  preferably  one  of  low  re- 
sistance, should  be  used.  A  convenient  form  of  thermoelement 
for  low  temperature  measurements  is  a  copper-constantan  couple 
which  gives  about  0.000017  volt  for  a  difference  of  temperature 
of  one  degree. 

FORM    OF   RECORD. 

E.rercise  65.     To  determine  the  thermoelectromotive  force  of 

a  thermoelement. 

Date .  . 


Fig.  98. 


Galvanometer  

Volt    sensitiveness.... 
Resistance  in  circuit.  . 


Temperature 


Deflection 


E.  M.  F. 


ELECTROMOTIVE    FORCE   AND   INTERNAL   RESISTANCE  OF   BATTERIES. 

135.  Electromotive  Force  and  Potential  Difference.  When- 
ever a  battery  is  closed  through  a  conductor  and  no  external  work 
is  done  except  that  of  heating  the  conductor,  we  have  by  Kirch- 
hoff's  second  law 

,  (188) 


174 


PHYSICAL    MEASUREMENTS 


where  H  is  the  electromotive  force  of  the  battery,  /  the  current, 
R  the  external  resistance  and  r  the  internal  resistance  of  the  bat- 
tery. In  case  polarization  occurs,  the  counter  E.  M.  F.  of  polari- 
zation must  be  subtracted  from  H.  According  to  the  above  form- 
ula the  drop  of  potential  in  the  whole  circuit  may  be  considered 
as  divided  into  two  parts:  (a),  I  R,  the  drop  of  potential  over  the 
external  resistance  commonly  called  the  terminal  potential  dif- 
ference, and  (b),  I  r,  the  drop  in  potential  in  the  cell  itself.  The 
relative  potentials  of  the  various  parts  of  the  circuit  may  be  rep- 
resented as  in  Fig.  99,  where  the  potentials  are  given  as  ordinates 
and  the  resistances  as  abscissae. 


Fig.  99. 


Consider  the  case  of  the  Daniell  cell.    The  potential  of  the  zinc 
(being  the  lowest  in  the  whole  circuit)  is  represented  by  the  point 

A.  Between  the  Zn  plate  and  the  ZnSO4  solution  there  is  a  sud- 
den rise  in  .potential  of  0.52  volt,  so  that  the  potential  of  the 
ZnSO4  solution  next  to  the  Zn  plate  is  represented  by  the  point 

B,  Then  follows  a  drop  in  potential  in  the  cell  until  the  Cu  elec- 
trode is  reached  when  a  second  sudden  rise  of  0.58  volt  occurs, 
the  potential  of  the  CuSO4  next  to  the  copper  plate,  and  that  of 
the  Cu  electrode  being  represented  by  the  points  C  and  D  respect- 
ively.    From  the  copper  plate,  the  potential  falls  off  regularly 
owing  to  the  external  resistance  R  until  the  zinc  plate  is  again 
reached  at  A\ 

To  complete  the  analogy,  the  figure  should  be  considered  as 
wound  upon  the  surface  of  a  cylinder  so  as  to  make  A'  coincide 


POTENTIAL  DIFFERENCE  1/5 

with  A.  If  both  solutions  are  normal  there  is  no  appreciable  dif- 
ference of  potential  between  the  liquids  themselves.  The  electro- 
motive force  £  of  a  cell  is  the  sum  of  the  potential  differences 
at  the  plates,  i.  e.,  AB  +  CD  =AG,  while  the  value  of  the  ter- 
minal potential  difference  £',  is  represented  by  FD,  and  depends 
upon  the  relative  values  of  the  external  and  internal  resistances, 
R  and  r.  In  general  this  relation  may  be  expressed  thus : 

E:E'::R  +  r:R  (189) 

whence 

r  =  R  .   ^— £,-.  (190) 

±i 

From  the  above  equations  it  is  evident  that  E'  equals  £  only 
when  R  is  infinitely  great,  and  E'  is  zero  for  R  equal  to  o.  In 
measuring  the  E.  M.  F.  of  a  battery  by  means  of  a  voltmeter  it 
must  be  noted  that  the  voltmeter  measures  only  the  difference  of 
potential  at  its  terminals,  and  that  its  resistance  must  be  very  large 
in  comparison  with  r,  if  it  is  to  give  reliable  values  for  £. 

136.  Exercise  66.  Terminal  Potential  Difference  as  a  Func- 
tion of  the  External  Resistance.  Con- 
nect a  voltmeter  or  galvanometer  of 
high  resistance  to  the  terminals  of  a 
Daniell  cell  B,  as  shown  in  Fig.  100. 
The  reading  will  give  practically  the 
E.  M.  F.  of  the  cell  or  a  deflection 
which  is  proportional  to  it.  Close  the 
circuit  through  R  by  means  of  the  key 
K.  The  voltmeter  will  now  read  F/.  **•  I0°- 

Vary  the  parallel  resistance  R,  by  steps  as  follows :  60,  40,  20,  10, 
6,  4,  2,  i,  and  0.5  ohms,  and  observe  the  corresponding  values  of 
£'.  Plot  E'  and  R  as  shown  in  Fig.  101.  The  curve  will  be  an 
hyperbola,  if  r  remains  constant.  Why?  The  resistance  r  of  the 
cell  may  be  found  directly  from  the  curve,  for  r  =  R  when 
£'  — £/2.  Calculate  also  the  value  of  r  from  equation  (190), 
and  plot  r  as  a  function  of  /  which  equals  E' /R. 

If  the  resistance  of  the  voltmeter  be  low,  the  external  resist- 
ance should  be  calculated  from  the  law  of  shunts. 


176 


PHYSICAL    MEASUREMENTS 


FORM    OF   RECORD. 

Exercise  66.    To  determine  the  terminal  potential  difference  of 
a  Daniell  cell  as  a  function  of  the  external  resistance. 


Voltmeter  No. 


Date. 


Resistance  box 
£  (for  R  infinite) 


le 

E' 

r 

I 

!).'! 

10 

B 

i 

s 

*OS 

a 

•H 

^oc 

•< 

§04 

s 

Q2 

0 

F 

M. 

F 

73 
^ 

jal    nifference^ 

^j—-" 

eiil 

0  

^ 

1 

/^ 

~~ 

-o— 

>  — 

_o_: 

/^ 

25 

a/ 

Jlesisi 

lan( 

?e 

4 

/ 

—  O 

/ 

Daniell  Cell 

7 

02.T 

oso  Current 

1/2 

Amps. 

ig 

175 

200 

External     Resistance 

Fig.  101. 

137.  Exercise  67.  Electromotive  Force  and  Internal  Re- 
sistance by  Voltmeter  and  Ammeter.  This  method  is  a  modifi- 
cation of  the  preceding.  A  milammeter  is  joined  in  series  with 
the  resistance  box  R  in  Fig.  100.  The  reading  for  H  is  made  with 
the  key  open  and  then  readings  for  £'  and  /  are  taken  simultane- 
ously, with  K  closed. 
Since 


we  have 


(191) 


(192) 


Vary  the  resistance  of  the  ammeter  circuit  and  take  three  dif- 
ferent currents  for  each  cell,  but  all  of  such  a  value  that  E'  re- 
mains nearly  equal  to  £/2.  If  R  and  the  resistance  of  the  amme- 
ter be  known,  equation  (190)  may  be  used  as  a  check  formula, 
giving  check  values  r',  which  should  agree  with  those  of  r.  In 
the  case  of  dry  cells  the  current  is  frequently  so  small  that  it  can- 


ELECTRIC  CEUvS 


1/7 


not  be  read  accurately.  In  this  case  compute  r  from  the  values 
of  E,  £',  and  R.  Since  the  polarization  is  quite  rapid  in  some 
cells,  it  is  better  to  read  E  immediately  after  the  key  K  is  opened, 
than  at  the  beginning  of  each  observation.  The  key  is  closed  just 
long  enough  to  take  the  readings. 


Name  of  cell 

£ 

X   |/ 

R  +  R™ 

r 

/ 

X.  . 

... 

FORM    OF1   RECORD. 

Exercise  6j.  To  determine  electromotive  force  and  internal  re- 
sistance of  five  different  cells  by  voltmeter  and  ammeter. 

Voltmeter  No  ____ 
Ammeter  No 
Resistance  box 

138.  Exercise  68.  Electromotive  Force  and  Internal  Re- 
sistance by  Condenser  Method.  The  foregoing  method  has  two 
disadvantages  :  First,  the  voltmeter  reading  is  only  approxi- 
mately equal  to  the  E.  M.  F.  of  cells  of  high  internal  resistance, 
and  second,  the  key  K  must  remain  closed  until  the  readings  can 
be  taken,  thus  permitting  of  considerable  polarization.  In  the 
condenser  method  the  first  objection  is  entirely  overcome  and  the 
time  needed  for  the  observation  is  much  shorter  than  in  the  pre- 
vious method.  The  principle  of  this  method  is  to  obtain  a  deflec- 
tion of  a  ballistic  galvanometer  proportional  to  the  E.  M.  F.  or 
to  the  difference  of  potential  of  the  cell. 

Two  methods  may  be  employed,  (a)  As  shown  in  Fig.  102, 
a  condenser  of  capacity  C,  and  a  ballistic 
galvanometer  G  are  substituted  for  the 
voltmeter  in  Fig.  100.  When  the  key  K^ 
is  pressed  down  the  condenser  is  charged 
with  a  quantity  of  electricity  proportional 
to  the  difference  of  a  potential  at  the  ter- 
minals of  the  battery,  and  when  the  key 
swings  back  to  the  upper  contact  a  deflec- 
tion is  obtained  proportional  to  the  charge, 
i.  e.,  proportional  to  the  difference  of  po- 
tential of  the  battery  terminals.  In  order 
to  charge  the  condenser  with  the  E.  M.  F. 


Fig.  102. 
of  the  battery,  keep  key  Kz  open  and  operate  only  key  K 


To 


PHYSICAL    MEASUREMENTS 


obtain  deflections  proportional  to  the  difference  of  potential  £' 
on  closed  circuit,  close  K2,  charge  and  discharge  the  condenser 
and  immediately  afterwards  open  K2.  Since  the  discharge  is 
completed  in  a  small  fraction  of  a  second  it  is  not  necessary  to 
wait  for  the  completion  of  the  swing  of  the  galvanometer  coil 
before  opening  K2. 

A  "pendulum  apparatus"  which  automatically  operates  four 
keys,  closing  the  circuit,  charging  and  discharging  the  condenser 
and  finally  opening  the  circuit,  is  described  in  Carhart  and  Pat- 
terson's Electrical  Measurements,  pp.  107-109.  This  instrument 
is  very  useful  for  this  exercise,  not  alone  because  it  rapidly  per- 
forms the  four  operations  needed,  but  also  because  in  successive 
observations  always  the  same  time  interval  elapses  between  the 
closing  of  the  circuit  and  the  charging  of  the  condenser. 

(b)  The  apparatus  is  arranged  as  in  Fig.  103.  The  key  K 
is  a  sliding  contact  which  when  passing  from  a  to  c  makes  for  a 
short  time  connection  between  a  and  b  and  charges  the  condenser 

C  through  the  galvanometer, 
the  resulting  deflection  being 
proportional  to  the  difference 
of  potential  at  the  terminals  of 
the  battery.  To  obtain  a  de- 
flection proportional  to  £  key 
KI  should  be  open;  to  obtain 
a  deflection  with  £',  K^  is 
closed,  and  K  is  turned  from 
a  through  b  to  c.  This  closes 
the  circuit  as  soon  as  the  slider  connects  a  and  b  and  at  the  same 
time  the  condenser  is  charged.  As  the  slider  advances  the  gal- 
vanometer circuit  is  broken  and  shortly  after  also  the  current 
through  R.  Since  very  short  time  is  needed  to  establish  the  cur- 
rent through  R  and  charge  the  condenser  fully,  the  momentary 
contact  between  a  and  b  suffices.  It  is  evident  that  there  can  be 
only  slight  polarization  during  this  brief  interval.  The  condenser 
is  charged  at  the  instant  the  circuit  is  closed. 

KI  should  be  opened  before  the  slider  is  pushed  back  to  its  orig- 
inal position.  After  each  charge  the  condenser  must  be  discharged 


IQ3- 


ELECTRIC  CELLS 


179 


completely  by  the  closing  of  key  K2,  which  should  be  kept  closed 
for  a  short  time  to  remove  all  absorbed  charge  if  such  be  present. 
In  order  to  express  the  E.  M.  F.  in  volts,  the  condenser  is  also 
charged  from  a  standard  cell  and  discharged  through  the  gal- 
vanometer. Let  the  deflection  in  this  case  be  d2,  and  let  d  be  the 
deflection  given  by  the  cell  under  investigation  when  on  open 
circuit,  and  d^  when  the  cell  is  closed  through  the  resistance  R. 
The  deflections  are  proportional  to  the  quantities  of  electricity 
passing  through  the  galvanometer,  and  these  are  themselves  equal 
to  the  product  of  the  difference  of  potential  at  the  terminals  of  the 
condenser,  times  the  constant  capacity,  C.  Therefore  if  £8  be  the 
E.  M.  F.  of  the  standard  cell,  then 


and 


d, 


(193) 


(i94) 


It  should  be  observed  that  the  internal  resistance  of  batteries, 
and  especially  of  dry  batteries,  appears  not  to  be  a  constant,  but 
to  decrease  with  a  decrease  of  R,  that  is  with  increasing  current 
furnished  by  the  cell.  This  is  probably  due  to  polarization, 
whose  effect  is  neglected  in  the  formulae  given  above.1  Use  three 
different  resistances  for  each  cell,  such  that  d±  is  about  d/2. 


FORM   OF  RECORD. 


Exercise  68.    To  determine  electromotive  force  and  internal  re- 
sistance of  five  cells  by  the  condenser  method. 

Date. 


Condenser 

Name  of  cell 

d 

cf« 

R 

E 

r 

Resistance   

Deflection,   standard  cell 

Temp   of  standard  cell 

E.  M.  F.  of  standard  cell.. 

139.  Exercise  69.  Internal  Resistance  by  Method  of 
Nernst  and  Haagn.  In  this  method  the  internal  resistance  of 
the  cell  is  determined  by  means  of  an  alternating  current,  while 
the  cell  itself  furnishes  no  current  whatever.  In  this  way,  if  the 


1  Guthe,  Physical  Review,  VII,  p.  193,  1898. 


i8o 


PHYSICAL    MEASUREMENTS 


period  of  the  alternating  current  is  sufficiently  small,  the  disturb- 
ing effects  of  polarization  are  entirely  avoided.  The  alternating 
current  is  furnished  by  the  induction  coil  /.  C.  (Fig.  104),  which 
charges  the  condenser  C  alternately  to  positive  and  negative  poten- 
tials. These  alternations  are  transmitted  to  a  Wheatstone  bridge, 

of  which  the  condensers  CL  and  C<> 
form  two  arms,  and  the  noninductive 
resistances  R  and  r  form  the  other 
two,  where  r  is  the  resistance  of  the 
battery  B.  The  resistance  R  is  so  ad- 
justed as  to  produce  minimum  sound 
in  the  telephone  receiver  which  re- 
places the  galvanometer  in  the  ordi- 
nary Wheatstone  bridge. 

The  applied  difference  of  potential 
produces  a  current  i,  through  the  re- 
Fig-  I04-  sistances  R  and  r.    Condenser  C\  will 

be  charged  with  a  quantity  Cjr,  and  condenser  C2  with  a  quantity 
C.2iR.  For  a  minimum  sound  in  the  telephone  these  charges  are 
equal  or 

(195) 


or 


(196) 


In  practice  it  is  best  to  connect  a  slide-wire  bridge  in  series 
with  the  battery  and  connect  the  telephone  with  the  contact  maker. 
We  have  then  instead  of  r  in  one  arm  of  the  Wheatstone  bridge, 
r  -(-  R',  where  R'  is  the  resistance  of  the  part  of  the  wire  from 
the  battery  B  to  the  contact  maker.  The  resistance  of  the  other 
portion  of  the  slide  wire  forms  of  course  part  of  R.  The  formula 
becomes  in  this  case 

r  =  ^R-R'  (197) 

FORM    OF   RECORD. 

Exercise  6p.  To  determine  the  internal  resistance  of  five  cells 
by  the  method  of  Nernst  and  Haagn. 

Name  of  cell  d  C2  R  R' 


CURRENT  l8l 

MEASUREMENTS  OF  CURRENT. 

140.  Measurable  Effects  of  a  Current.    Since  it  is  obviously 
impossible  to  preserve  standards  of  current,  it  is  also  impossible 
to  determine  currents  by  direct  comparison  with  a  standard  as  in 
the  case  of  resistance  or  electromotive   force.     It  is   necessary 
therefore  to  employ  certain  known  effects  of  the  current  for  pur- 
poses of  measurement.     The  following  are  those  chiefly  used  in 
laboratory  practice,      (a)   Electromagnetic  effect  as  applied  in 
galvanometers,    voltmeters    and    ammeters.      These    instruments 
have  been  in  frequent  use  for  the  measurement  of  currents  in  the 
previous  experiments.     If  any  doubt  as  to  their  accuracy  arises 
they  should  be  calibrated  by  one  of  the  methods  which  follow. 
(&)     Chemical  effect,  as  applied  in  the  copper  or  silver  coulom- 
eter.      (c)   The  production  of  a  potential  difference  at  the  ter- 
minals of  a  known  resistance,  as  exemplified  in  Exercise  71. 

141.  Law  of  Electrolysis.     According  to  Faraday's  law1  the 
quantity  of  a  substance  deposited  by  an  electric  current  is  pro- 
portional to  the  quantity  of  electricity  passing  through  the  elec- 
trolytic cell.     A  steady  current  of  one  ampere  will  deposit  4.025 
grams  of  silver  or  1.1838  grams  of  copper  in  one  hour.    The  elec- 
trochemical equivalent  s,  of  a  substance  is  the  ratio  of  the  mass 
of     the    substance     deposited     to     the    quantity    of    electricity 
flowing     through     the     coulometer.       Since     the     quantity     of 
electricity  is  the  product  of  current  and  time  it  is  easy  to  deter- 
mine the  average  current  flowing  through  an  electrolytic  cell, 
by  dividing  the  number  of  grams  of  the  substance  deposited  in 
time  t  by  the  weight  which  would  have  been  deposited  by  unit 
current  in  the  same  time.    Thus  if  iv  be  the  number  of  grams  de- 
posited, then 

(198) 

Whenever  it  is  desired  to  calibrate  an  instrument  for  measur- 
ing current  by  means  of  the  coulometer,  it  is  necessary  to  deter- 

1  College  Physics,  Article  283. 


182 


PHYSICAL    MEASUREMENTS 


mine  the  average  reading  of  the  instrument  during  the  time  the 
current  flows,  and  by  comparing  this  reading  with  the  average 
current,  to  determine  the  constant  of  the  instrument  or  the  cor- 
rection to  be  applied  in  the  case  of  direct  reading  instruments. 

142.  The  Copper  Coulometer.  Of  the  various  forms  of 
coulometers  the  simplest  is  the  copper  coulometer  (Fig.  105).  Its 
manipulation  demands  considerable  care.  The  coulometer  con- 
sists of  two  copper  electrodes  in  the  form  of  plates  or  spiral  wires, 
immersed  in  a  solution  of  CuSO4.  This  solution  must  be  kept  in 
a  separate  bottle  and  be  used  for  this  experiment  only.1  That  part 
of  the  kathode  on  which  the  copper  is  to  be  deposited  must  be  kept 
perfectly  clean  and  must  not  be  touched  with  the  fingers.  If  not 
clean,  the  plate  must  be  dipped  into  a  strong  solution  of  potassium 
cyanide  then  washed  well  with  water  and  then  dipped  into  strong 
alcohol  and  dried.  All  these  operations  should  be  performed  as 
rapidly  as  possible  since  moist  copper  oxydizes  easily  in  the  air. 
In  case  the  kathode  is  perfectly  clean  it 
may  be  used  without  further  preparation. 

143.  Exercise  70.  Calibration  of  an 
Instrument  by  use  of  Copper  Coulom- 
eter. First  weigh  the  kathode  to  o.i  milli- 
gram. Then  set  up  the  coulometers,  two 
in  series,  in  order  to  check  the  result.  In 
the  circuit  place  a  battery  of  high,  and  if 
possible,  of  constant  E.  M.  F.,  a  large  vari- 
able resistance,  the  instrument  to  be  cali- 
brated, and  a  key  for  opening  and  closing 
the  circuit.  The  resistance  should  be  adjusted  beforehand  so  as  to 
give  the  current  that  is  to  be  sent  through  the  coulometers.  The 
current  density  may  be  as  high  as  1.5  amperes  per  square  deci- 
meter of  the  electrodes.  Close  the  circuit  for  a  definite  time,  say 
thirty  minutes,  and  note  the  reading  of  the  instrument  every  min- 
ute. After  the  circuit  is  opened  wash  the  kathode  in  plenty  of 
water,  rinse  in  alcohol,  dry  and  weigh  to  determine  the  mass  of 


Fig.  105. 


1Tihe  following  solution  is  recommended:    CtiSO*,  15  grams; 
5  grams;  alcohol,  5  grams;  water,  100  grams. 


CURRENT  183 

the  deposit.    Apply  equation  (198)  for  the  calculation  of  the  cur- 
rent. 

FORM   OF  RECORD. 

Exercise  jo.     To  calibrate  a by  copper  Goniometer. 

Date 

Name  and  number  of  instrument....       Formula  of  instrument.. 


Wt.  of  kathode  before 
Wt.  of  kathode  after 

Gain 

Average  gain 

Average  current 


Coulometer  I 

Coulometer  II 

1 

Reading 

" 

Average  reading  of  Inst 

Constant  (or  correction)  of  Inst.. 


144.  Exercise  71.  Calibration  of  Ammeter  by  Standard 
Cell.  A  current  may  be  measured  by  comparing  the  difference  of 
potential  at  the  terminals  of  a  known  resistance  r,  through  which 
it  flows,  with  a  known  E.  M.  F.  Let  A  (Fig.  106),  be  an  ammeter 
to  be  calibrated ;  B  a  battery  of  constant  E.  M.  F.,  and  j  an  adjust- 
able resistance  for  varying  the  current.  Ri  and  R2  are  two  resist- 
ances, each  very  large  in  comparison  with  r.  The  standard  cell 


is  set  in  such  a  way  as  to  oppose  the 
difference  of  potential  at  the  terminals 
of  Rif  The  resistances  R-L  and  R2  are 
so  adjusted  that  on  closing  K'  after 
K,  no  current  will  flow  through  the 
galvanometer;  then 


(199) 


where  I  is  the  current  flowing  through 
r,  and  i  that  through  J?±  and  R*.    I  is 
sensibly    the    same    current    as    that 
through  the  ammeter. 
Moreover 

£8  =  iRi 
therefore 


St.C. 


(201) 


It  is  evident  that  under  the  conditions  given  the  current  must 


1 84 


PHYSICAL    MEASUREMENTS 


be  equal  to  or  greater  than  Es/r,  in  order  to  make  a  balance  possi- 
ble.1   Calibrate  an  ammeter  using  at  least  five  different  currents. 


FORM   OF   RECORD. 

Exercise  71.    To  calibrate  ammeter  No 

Date 

Resistance  r. 
Temperature.  ...      E.  M.  F. . . . 


Zero  point  of  ammeter. . . . 
Standard  cell  No.. 


.A 
Observed 

.mmeter  reading 
Corrected  for  zero  pt. 

ff, 

R, 

I  computed 

Correction 

COMPARISON  OF  CAPACITIES. 

145.     Exercise  72.     Comparison  by  Direct  Deflection.    The 

simplest  method  of  determining  the  capacity  of  a  condenser  con- 
sists in  comparing  the  deflections  of  a  ballistic  galvanometer 
(Fig.  74),  caused  by  the  discharge  of  the  quantities  of  electricity 
stored  in  a  standard  condenser  and  in  that  under  investigation, 
when  each  has  been  charged  to  the  same  difference  of  potential. 
From  Exercise  50, 


=  QJc  = 


EC, 


(202) 


(203) 


(204) 


One  of  the  condensers  Clt  is  a  standard  condenser  of  known 
capacity. 


xFor  a  method  for  measuring  small  currents  see  Carhart  and  Patter- 
son, Electrical  Measurements,  p.  172. 


CAPACITY 


If  the  capacity  of  a  condenser  is  so  large  as  to  give  too  large 
deflections,  when  discharged  through 
the  galvanometer,  the  experiment  may 
be  arranged  as  shown  in  Fig.  107.    A 
battery  of  constant  E.  M.  F.  sends  a 
current    through    two    resistances   J^t 
and  R2)  the  sum  of  which  must  be  kept 
constant.    The  condenser  is  connected 
over  one  of  these  resistances  R^,  and 
charged  with  the  difference  of  poten- 
tial at  its  terminals  equal  to  i  Rlt     If  Fi&-  I07- 
the  deflection  on  discharging  is  too  large  the  value  of  R±  is  to  be 
reduced,  R.2  being  increased  by  the  same  amount.    Then  if  C±  and 
C2  represent,  respectively,,  the  standard  and  the  unknown  capacity 


-i 

^4^ 

Rz 

\      R,      - 

i  R  \Ci          .    ,        i 

and  a2  = 


therefore 


(205) 
(206) 


Instead  of  two  resistance  boxes  the  wire  of  a  slide  wire  bridge 
may  be  used. 

Not  infrequently  condensers  are  found  whose  capacity  depends 
very  largely  upon  the  time  of  charge.  Such  condensers  are 
termed  absorbing  condensers.  The  quantity  of  electricity  stored 
up  in  such  condensers  may  be  considered  as  consisting  of  two 
parts  (a)  the  free  charge,  (&)  the  absorbed  charge.  In  discharg- 
ing through  a  galvanometer  the  free  charge  will  pass  first  and  then 
the  absorbed  charge  will  follow.  Part  of  the  latter  will  affect  the 
throw  of  the  galvanometer  and  the  "effective"  capacity  will  there- 
fore depend  to  a  certain  degree  upon  the  period  of  the  galvanom- 
eter, being  the  larger  the  larger  the  period.  To  obtain  results 
independent  of  the  period  the  discharging  circuit  must  be  opened 
a  few  hundredths  of  a  second  after  contact  with  the  galvanometer 
is  made.1  Thus  we  measure  only  the  free  charge  which  in  absorb- 


Zeleny,  Physical  Review,  Vol.  22,  p.  65,  1906. 


i86 


PHYSICAL    MEASUREMENTS 


ing  condensers  is  frequently  less  than  one  half  of  the  quantity  of 
electricity  observed  by  applying  the  above  method  after  long  con- 
tinued charging.  To  study  the  effect  of  absorption  the  student 
should  take  different  intervals  for  charging  and  note  carefully  the 
resulting  effect  upon  the  observed  capacity.  • 

FORM   OF  RECORD. 

Exercise  72.    To  measure  the  capacities  of  four  condensers. 

Date. . 


Resistance  boxes  .  . 

Cell.. 

I 

146.     Exercise  73.     Method  of  Mixtures.     The  apparatus  is 
arranged  as  shown  in  Fig.  108.     In  the  middle  of  the  figure  is 


Fig.  1 08. 

shown  a  double  throw  switch,  by  means  of  which  the  points  a 
and  of  may  be  connected  either  to  b  and  bf  or  to  c  and  c' .    By. 


means   of  the  first  connections   the  condensers 


and   C    are 


charged  with  quantities  of  electricity  equal  to  RiiCl  and'R2iC2  re- 


INDUCTANCE 


i87 


spectively.  On  reversing  the  switch  these  quantities  mix  and 
there  will,  in  general,  be  a  deflection  of  the  galvanometer  result- 
ing from  the  difference  of  these  two  charges  on  pressing  the  key 
k.  Adjust  the  resistances  R^  and  R2  until  there  is  no  deflection 
of  the  galvanometer ;  then 


and 


Ri  i  d  —  R2  i  C2 
z  =  C\  .  -r>— 


(207) 
(208) 


Name  of  condenser 

Time 

R\ 

R\ 

c 

sees 

ienser.  . 

The  battery  should  have  a  relatively  high  E.  M.  F. ;  six  to  ten 
Leclanche  elements  may  be  used.  The  disturbing  influence  of 
absorption  may  be  studied  as  in  the  preceding  experiment. 

FORM   OF  RECORD. 

Exercise  75.  To  compare  two  condensers  by  the  method  of 
mixtures. 

Galvanometer 
Resistance  'boxes 


MEASUREMENT  OF   INDUCTANCE:. 

147.  Exercise  74.  Selfinductance  of  a  Coil  Compared  with 
a  Standard.  Arrange  the  apparatus  as  in  Fig.  109,  in  which  the 
two  selfinductances  form  each  one  arm 
of  a  Wheatstone  bridge.  Let  their  re- 
sistance and  selfinductance  be  R:  and 
Llf  R2,  and  L2  respectively.  The  re- 
maining two  arms  are  formed  by  the 
noninductive  resistances  R?i  and  Rt. 
First  adjust  R3  and  R4  for  constant 
current  until  the  galvanometer  shows 
no  deflection  on  closing  k,  that  is, 


Fig.  109. 


R* 


(209) 


Then,  keeping  k  closed,  open  K.    There  will  in  general  be  a  de- 


l88  PHYSICAL    MEASUREMENTS 

flection  of  the  galvanometer  due  to  the  difference  of  the  E.  M.  F. 
produced  by  the  selfinductances  in  the  branches  Z^  and  L.2.  Vary 
the  selfinductance  of  the  standard,  until  upon  opening  K  no  de- 
flection is  obtained.  Then  the  points  connected  to  the  galvanom- 
eter will  be  at  the  same  potential  as  well  during  the  steady  as 
during  the  variable  state  of  the  current. 

Let  the  difference  of  potential  over  R^  or  R2  for  a  constant  cur- 
rent be  B ;  then  the  current  through  R^  is  ii  =  'E/R1,  and  through 
R2J  i,  =  n/R2.  Now  the  value  of  the  E.  M.  F.  due  to  selfinduc- 
tance is1 

'=-*•£•  (2IO) 

In  the  case  under  consideration,  the  value  of  e  on  opening  K  at 
any  moment,  is 


and 

<•=-•&••  sr=-j$r  (2I2) 

But  after  the  adjustment  of  the  standard,  e±  equals  e2  and 
dH/dt  is  the  same  for  both  e^  and  e.2,  and  hence  the  condition  for 
no  deflection  during -the  variable  state  of  the  current  is  given  by 
the  equation 

~R~  ~  ~R~  or  ^x  =  "k*  ~R~  =  ^  ~R~  ' 

In  practice  care  must  be  taken  to  place  the  two  selfinductances 
in  such  a  position  that  they  will  not  influence  each  other.  To  ob- 
tain accurate  results  the  battery  must  possess  a  high  voltage.  The 
best  results  are  obtained  by  using  an  alternating  current  and  sub- 
stituting for  the  galvanometer  a  telephone  receiver  as  in  Exercise 
6 1,  or  by  rectifying  the  current  before  sending  it  through  the  gal- 
vanometer. In  an  instrument  designed  by  Ayrton  and  Perry-, 
called  the  secohmmeter,2  this  rectification  of  the  current  is  ef- 
fected by  means  of  a  double  commutator. 


J  College  Physics,  Article  333. 

2  Carhart  and  Patterson,  Electrical  Measurements,  p.   no. 


INDUCTANCE  189 

FORM   OF  RECORD. 

Exercise  74.    To  measure  the  selfinductance  of 


Apparatus 
Voltage  of  battery 


148.  Exercise  75.  Mutual  Inductance  of  two  Coils.  Accord- 
ing to  definition  the  coefficient  of  mutual  induction  or,  the 
mutual  inductance  of  two  coils  is  given  by  the  equation1 


(214) 


Date  

R3 

ft 

Reading   of    standard 

L 

where  e  is  the  counter  E.  M.  F.  in  the  secondary  due  to  the  mutual 
inductance.  M,  and  i,  the  current  in  the  primary.  Let  the  induced 
E.  M.  F.  produce  a  current  through  a  ballistic  galvanometer  whose 
resistance  is  g  ;  the  current  i'  '  ,  so  produced,  will  at  any  moment  be 

,  where  r  +  g  is  the  total  resistance  of  the  secondary  circuit. 


The  total  quantity  of  electricity  passing  through  the  galvanometer 


s 


where  the  integral  is  to  be  taken  between  o  and  /,  the  final  value 
of  the  steady  current  in  the  primary  circuit.  Therefore  the  total 
quantity  in  the  secondary  circuit  when  the  primary  circuit  is 
closed,  becomes,  neglecting  its  direction 


If  the  current  through  the  primary  be  suddenly  reversed  the  value 
of  Q  is  doubled.  From  the  formula  for  the  ballistic  galvanometer 
we  have  Q  =  cd,  where  c  is  the  constant  of  the  galvanometer. 
Therefore 


1  College  Physics,  Article  332. 


igO  PHYSICAL    MEASUREMENTS 

This  formula  is  based  upon  the  assumption  that  no  counter 
electromotive  force  is  generated  in  the  galvanometer.  But  this 
assumption  is  not  correct,  especially  in  the  case  of  instruments  of 
the  d'Arsonval  type.  In  this  case  a  counter  electromotive  force 
is  set  up,  due  to  the  cutting  of  lines  of  magnetic  induction  by  the 
coil  swinging  in  the  field  of  the  permanent  magnet.  Since  the 
total  number  of  lines  cut,  or  the  change  of  flux  through  the  coil 
is  nearly  proportional  to  the  actual  deflection  d,  the  resulting 

quantity  of  electricity1  is  nearly  equal  to  —  -  ,  and  tending  to 

flow  in  the  opposite  direction,  would,  if  acting  alone  on  the  cir- 
cuit, produce  a  deflection  —  d2,  in  the  opposite  direction  from  d, 
while  if  there  were  no  inductive  effect  in  the  coil,  the  deflection 
would  be  simplv 


We  obtain,  therefore,  for  the  actual  deflection 


^-  (2I9) 


In  order  to  obtain  accurate  results  we  must  therefore  add  the 
quantity  k/c  to  the  actual  resistance  r  +  g  ;  in  other  words,  we 
must  here  use  the  apparent  resistance  of  the  galvanometer, 


k 

'  —  g  -|  --  .  (220) 


instead  of  its  ohmic  resistance  g. 
The  expression  for  M  thus  becomes 


(22I) 


The  apparent  resistance  g'  ,  of  the  galvanometer  may  be  deter- 
mined in  the  following  manner.  As  described  above,  produce  a 
deflection  of  the  galvanometer,  and  note  the  deflection.  Let  the 
total  resistance  of  the  secondary  circuit  be  r  +  g,  where  r  denotes 


1  College  Physics,  Article  330. 


INDUCTANCE 


191 


the  resistance  outside  the  galvanometer.  Next  introduce  in  series 
a  resistance  r' ' ,  such  that  with  the  same  primary  current  the  de- 
flection is  now  reduced  to  one  half  its  former  value.  Then 


c(r-\-g'}d  _ 
I 


and 


(222) 


The  deflection  is  proportional  to  the  current  in  the  primary 
and  inversely  proportional  to  the  resistance  of  the  secondary  cir- 
:uit.  To  show  these  relations 
dearly  make  the  following 
experiments:  The  arrange- 
ment is  shown  in  Fig.  no. 
Let  P  represent  the  primary 
and  £  the  secondary  coil. 
The  primary  is  in  series 
with  a  battery  B,  a  variable 
resistance  R  and  an  ammeter 
A.  To  the  secondary  coil  are 
joined  a  resistance  r  and  a 
ballistic  galvanometer. 

(1)  Vary  /  keeping  r  constant. 

(2)  Vary  r  keeping  /  constant, 
curves  should  be  straight  lines. 


Fig.  no. 

Plot  values  of  d  and  /. 
Plot  d  and  i/(r  +  g')  ;  both 


FORM   OF   RECORD. 

Exercise  75.    To  determine  the  mutual  inductance  of, 

Date, 


Ammeter  

(I] 
/ 

Bal 
)    r  co 
d 

listic  gi 
nstant 
M 

ilvanon 

(2 

r 

ieter  

Determination 
Capacity     .  . 

of  constant 

)    /    CO 

d 

istant 
M 

Electromotive 
Deflection 

force 

Constant    . 

Apparent    resistance  : 
r  .                  di 

r'..                d« 

a'  .  . 

CHAPTER  IX. 

MAGNETIC  MEASUREMENTS. 
MAGNETIC 


149.  Magnetic  Fields.  In  most  of  the  foregoing  experiments 
the  action  between  a  magnetic  field  produced  by  a  current  and 
another  magnetic  field  was  used  to  determine  electrical  quantities. 
In  the  following  experiments  magnetic  fields  and  their  effect  upon 
the  magnetic  properties  of  iron  and  steel  will  be  studied.  A 
magnetic  field  is  perfectly  determined  if  at  every  point  the  in- 
tensity H  of  the  field  and  the  magnetic  induction  B  =  ^  H  of  the 
field  be  known.  The  intensity  may  be  compared  to  a  stress  in  an 
elastic  body,  the  induction  to  a  strain1.  In  the  case  of  air,  since 
/A  equals  unity,  these  two  quantities  are  numerically  equal,  but  it 
should  be  kept  in  mind  that  they  are  different  physical  quantities. 

The  intensity  of  a  magnetic  field  at  a  given  point  may  be  meas- 
ured by  the  force  per  unit  pole  strength  acting  on  a  pole  placed 
at  this  point.  The  unit  of  intensity  of  magnetic  field  is  the  dyne 
per  unit  pole  and  is  called  the  gauss.  If  the  magnetic 
field  be  produced  by  an  electric  current  it  is  best  to  calculate  the 
intensity  at  any  point  from  the  current.  In  the  case  of  a  circular 
current  of  radius  r  it  is2 


H  =       y    /  gauss  (223) 

In  the  case  of  a  very  long  solenoid  it  iss 

H  =  4^  1  §auss  (224) 

where  n'  is  the  number  of  turns  of  wire  per  unit  length  of  the 
solenoid.     In  both  equations  7  is  expressed  in  amperes. 


1  College  Physics,  Article  242. 

2  College  Physics,  Article  257. 
8  College  Physics,  Article  319. 


MAGNETIC  FIELDS  193 

The  magnetic  induction  B  at  any  point  is  measured  by  the 
number  of  lines  of  induction  per  unit  crosssection  and  its  unit  is 
therefore  the  line  per  square  centimeter.  The  magnetic  flux  $ 
through  a  given  area  is  simply  the  number  of  lines  of  induction 
through  this  area  A,  or1 


(225) 


150.     Exercise  76.     Determination  of  H.     (First  Method). 

The  lines  of  magnetic  induction  due  to  the  earth's  field  run  from 
South  to  North,  although  deviating  in  some  places  by  an  ap- 
preciable angle  from  the  geographical  North  and  South  line.  This 
angle  is  called  the  angle  of  declination.  The  lines  of  induction  are 
also  inclined  towards  the  horizontal  plane,  making  in  Ann  Arbor 
an  angle  of  72°  with  the  horizon.  This  is  the  angle  of  magnetic 
inclination  or  the  magnetic  dip.  It  should  be  noted  that  neither 
the  magnetic  declination  nor  the  dip  is  constant.  They  not  only 
vary  from  place  to  place  over  the  earth's  surface,  but  they  also 
vary  slightly  from  year  to  year  and  from  day  to  day  in  the  same 
place.  The  magnetic  field  of  the  earth  may  be  conceived  as  the 
resultant  of  two  components,  one  horizontal  H,  and  one  vertical 
V.  Then  ©,  the  angle  of  dip,  is  given  by  the  equation 


tan  ®=  77--  (226) 

ri 


It  is  of  great  interest  to  determine  the  value  of  the  horizontal 
component  of  the  earth's  magnetism  in  terms  of  the  fundamental 
units  of  length,  mass  and  time,  since  all  the  practical  magnetic 
and  electrical  units  are  based  on  these2. 

A  relation  between  the  magnetic  moment  of    a  magnet  and  the 


1  College  Physics,  Article  324. 

2  College  Physics,  Articles  258  and  404. 


IQ4  PHYSICAL    MEASUREMENTS 

strength  of  the  magnetic  field  in  which  it  is  situated  may  be  de- 
rived in  either  one  of  two  ways : 

( i )    Method  of  deflections. 
Let  a  magnet  NS,  (Fig.  in) 
be    placed    with    its    axis    on 
the  magnetic  East- West  line1 
and  on  this  line  in  the  same 
horizontal   plane,   at   a   meas- 
]JV     ured  distance  from  it  place  a 
very    small    magnetic    needle 
ns,   suspended  by   a  fine   silk 
Fig.  in.  thread  or  quartz  fiber. 

Let  m  and  /  be  the  pole  strength  and  half-length  of  magnet  NS.    Let 
*  m'  and  •/*  represent  the  same  for  magnet  n  s. 

Also  let 

d  'be  the  distance  between  the  centers  of  the  magnets, 
H  the  horizontal  component  of  the  earth's  magnetic  field, 
M  the  magnetic  moment  of  the  magnet  NS  =  2ml, 
M'  the  magnetic  moment  of  the  magnet  n  s  =  2m'/'. 

The  magnet  ns  will  be  deflected  from  its  normal  position  by 
the  angle  <f>  such  that  the  turning  moments  due  to  the  earth's  field 
and  that  due  to  the  influence  of  the  magnet  NS  are  equal2.  The 
force  due  to  the  earth's  field  on  one  of  the  poles  of  ns  is  m'H, 
and  the  lever  arm  on  which  it  acts  is  /'  sin  </>;  so  the  turning 
moment  on  the  whole  magnet  is 

2m'l'H  sin  <f>  =  M'H  sin  <£  .  (227) 

The  force  on  the  north-seeking  pole  of  ns  due  to  the  south- 
seeking  pole  of  NS  is,  according  to  Coulomb's  law  of  the  inverse 

squares  --  (Jnm,.2   ,  I'  being  considered  negligible  in  comparison 

1  To  find  the  magnetic  East-West  line,  suspend  in  the  center  of  a  plane 
coil  of  wire  a  magnetic  needle.    Turn  the  coil  until  on  sending  a  current 
through  it  the  needle  is  not  deflected.    The  plane  of  the  coil  is  then  in  the 
magnetic  East-West  line. 

2  The  angle  <t>  is  determined  by  mirror  and  scale.    Suppose  the  distance 
of  the  mirror   from    the  scale    to    be    D    and    the    deflection    d,    then 
tan  2$ 


MAGNETIC  FIELDS  195 

with  d.     The  force  of  the  north-seeking  pole  of  NS  is  in  the 


opposite  direction  and  equal  to  —     77^      j  so  that  the  whole  force 
on  the  north-seeking  pole  of  the  needle  is  : 


D        m  m  x  i 

1  =    ~  '  L(d-/)a  *~  (d  +  01  J  (       } 

and  if  /  be  small  in  comparison  with  d,  and  since  p.  is  unity, 

(229) 

The  turning  moment  due  to  this  force  is  FJ'  cos  <j>,  and  since 
the  turning  moments  on  the  two  poles  of  ns  are  equal  and  in  the 
same  direction  the  total  turning  moment  exerted  on  ns  by  the 
magnet  NS  is  2  FJ'  cos  <f>  or 

8mm'//'  MM' 

7P COS0^=2  . — -p —  COS  0  .  (23O) 

But  the  turning  moments  produced  by  the  forces  exerted  by 
the  earth  and  the  magnet  NS  must  be  equal  since  the  needle  is  in 
equilibrium.  So  the  expressions  for  these  turning  moments  may 
be  set  equal  to  each  other,  or 

MM' 
M  Hsm<t>  =  2  —£ —  cos0  (231) 

whence 

M        d3 

—  =  —.  tan  </>  =  A.  (232) 

(2)  Method  of  oscillations.  The  law  of  vibration  of  a  mag- 
netic needle  in  a  magnetic  field  is  the  same  as  that  of  the  physical 
pendulum.  If  we  suspend  the  deflecting  magnet  NS,  so  as  to 
swing  freely,  its  period  is 

IK 


where  T  is  the  period  of  a  complete  vibration  and  K  is  the  mo- 
ment of  inertia  of  the  magnet.       MH  is  the  torque  when  the 


PHYSICAL    MEASUREMENTS 


magnet  is  at  right  angles  to  the  force.  In  order  to  find  K  the 
same  method  is  applied  as  in  Exercise  26,  by  putting  a  brass  ring 
of  known  moment  of  inertia  K'  on  the  magnet  so  that  the  axis  of 
rotation  and  the  axis  of  the  ring  coincide.  Let  the  period  of 
vibration  of  the  new  system  be  T',  then 


and  from  the  equations  (233)  and  (234) 

K' 


(235) 


Equations  (232)  and  (235)  furnish  two  expressions  involving 
M  and  H. 

•%•=*  (236) 

and 

MH  —  B, 
from  which 


or  on  substituting  the  values  of  A  and  B, 


It  is  obvious  that  by  this  method  the  value  of  H  is  deter- 
mined in  the  fundamental  units  of  mass,  length  and  time. 

The  instrument  used  for  this  experiment  is  called  a  magnet- 
ometer. It  consists  of  a  closed  case  furnished  with  windows 
to  enable  the  observer  to  measure  the  deflection  of  the  needle  by 
means  of  the  mirror  which  it  carries.  Place  the  deflecting  magnet 
at  a  certain  distance  from  the  small  needle  as  described  above 
and  observe  the  deflection.  Turn  the  magnet  end  for  end  and 
again  observe  the  deflection,  which  will  now  be  in  the  opposite 
direction.  Repeat  these  two  operations  with  the  magnet  at  the 


MAGNETIC 


197 


same  distance  on  the  opposite  side  of  the  needle  and  take  the  mean 
of  the  four  observations  as  the  deflection  <£.  In  case  the  length 
of  the  deflecting  magnet  is  not  negligible  in  comparison  with  the 
distance  d,  we  may  take  it  into  account.  Kohlrausch  has  shown 
that  for  a  bar  magnet  the  distance  between  the  poles  is  very  nearly 
5/6  of  the  length  of  the  magnet  and  this  value  should  be  used  for 
2/.  The  formula  for  H  becomes 


2TT 


2K'd 


(239) 


In  the  vibration  method  the  time  of  vibration  may  be  determined 
as  in  Exercise  13,  or  by  the  use  of  an  ordinary  stop  watch  if  less 
accuracy  is  required.  The  stop  watch  should  however  be  com- 
pared with  a  standard  clock.  How  may  the  magnetic  moment  of 
the  deflecting  magnet  be  determined  from  the  foregoing  formulae? 

FORM   OF   RECORD. 

Exercise  76.     Determination  of  H. 


\  L  J 

Determination  of  -77 
H 

Distance    d 

Position  of 
magnet. 
d 

Deflection 

tan  2  0 

tan  0 

L/ensf'th  of  magnet 

Reversed 

b 

Reversed 

Computation  of 


;r 


(2)    Determination  of  M H: 

(a)  Determination  of  T. 
(&)  Determination  of  T'. 
(c)  Data  for  moment  of  inertia  of  ring. 

'Mass  of  ring Outer  diameter  of  ring. 

Moment  of  inertia  of  ring 

Computation  of  MH. 

3)  Computation  of  H. 


Inner. 


198  PHYSICAL    MEASUREMENTS 

151.     Exercise  77.     Determination  of  H.  (Second  Method). 

In  formula   (233), 

T  —  2. 


the  torsional  moment  of  the  suspending  wire  was  neglected.    The 
complete  formula  for  a  given  magnet  is 


or  (240) 


where  <3\  is  the  moment  of  the  torsional  couple  of  the  suspending 
wire,  and  c  and  c'  are  constants.  These  relations  are  illustrated 
in  the  following  experiment.  Let  a  short  magnetic  needle  be  sus- 
pended at  the  center  of  a  solenoid  whose  length  is  at  least  twenty 
times  its  diameter,  and  whose  axis  is  parallel  to  the  magnetic 
meridian.  Observe  the  period  of  vibration  of  the  needle,  first 
when  swinging  in  the  earth's  field  H,  and  then  when  a  magnetic 
field  Hlf  due  to  a  known  current  /  through  the  soleniod,  is  super- 
posed upon  the  field  of  the  earth.  Let  the  period  of  vibration  of 
the  system  in  the  first  case  be  T  and  in  the  second  T±.  Then 


(241) 
(242) 


Vary  the  current  and  observe  Tit  Tz,  Ts,  T±,  etc.  ;  also  the 
periods  of  vibration  T\,  T'2,  T's,  etc.,  when  the  current  through 
the  solenoid  is  reversed.  Then  plot  i/T2  as  ordinates  attd  the  mag- 
netic field  strength  produced  by  the  currents  as  abscissae,  the  latter 
positive  or  negative  according  to  its  direction1.  The  curve  consists 


1lf  the  current  be  increased  in  the  negative  sense  beyond  a  certain 
value  tfhe  magnet  turns  through  180°.    Why? 


MAGNETIC 


199 


of  two  straight  lines  meeting  at  the  point  0' ',  Fig.  112.  The  time 
corresponding  to  the  ordinate  0  B  is  then  the  period  of  vibration 
corresponding  to  the  rigidity  of  the  suspension  alone.  If  0'  be 
considered  as  the  origin  and  the  strength  of  field  be  plotted  on 
O'x  it  is  evident  that  O'B  is  the  value  of  the  horizontal  component 
of  the  earth's  field,  and  that  the  strength  H'  of  any  field  may  be 
readily  determined  by  allowing  the  magnet  under  experiment  to 
vibrate  in  this  field  and  determining  its  period  of  vibration.  Then 


H' 


(243) 


where  c  and  c'  are  to  be  substituted  from  the  foregoing  observa- 
tions. 


A 


X 


_.3V 


-ff=Q19 

--LI-I- 


£ 


Strength    of    Magnetic    Field 

Fig.  112. 

In  practice  a  storage  cell  of  constant  E.  M.  F.  is  used  to  furnish 
the  current  through  the  solenoid.  A  resistance  is  joined  in  series 
to  allow  the  current  to  be  varied  within  wide  limits.  The  periods 
of  vibration  may  be  determined  by  means  of  a  stop  watch.  From 
equation  (224)  the  strength  of  field  at  the  center  of  the  solenoid 


s 


•*  •*•  1    — "  T 

10  L 

where  N  is  the  number  of  turns  of  wire  and  L  the  length  of  the 
solenoid.  The  current  /  is  computed  from  the  electromotive  force 
of  the  cell  and  the  total  resistance. 


2OO 


PHYSICAL    MEASUREMENTS 


FORM    OF   RECORD. 


Exercise  77.     Determination  of  H.     (Second  Method). 


Dimensions 

of   solenoid  : 

R 

I 

T 

r 

n  — 

Determined 

from  curve  ° 

c  — 

ZJr  

MAGNETIC  PROPERTIES  OF  IRON  AND  STEEL. 

152.  Magnetizing  Field  and  Permeability.  When  a  bar  of 
unmagnetized  iron  or  steel  is  introduced  into  a  magnetic  field 
the  distribution  of  lines  of  magnetic  induction  is  greatly  altered. 
The  number  of  lines  per  square  centimeter  at  any  point  in  the  baj* 
is  considerably  larger  than  it  was  at  the  same  point  before  the 
introduction  of  the  iron.  In  other  words  B  which  is  called  the 
induction  is  much  increased.  At  the  same  time  the  intensity  of 
the  field  inside  the  iron  is,  in  general,  much  smaller  than  it  was 
before  at  that  place.  This  intensity  which  is  called  the  magnetiz- 
ing intensity  may  be  considered  as  being  the  result  of  the  super- 
position of  the  original  field  and  the  demagnetizing  effect  of 
the  ends  of  the  iron  bar.  This  effect  is  in  the  opposite  direction 
to  the  original  field  and  thus  the  intensity  of  the  field  is  reduced. 
The  demagnetizing  effect  of  the  ends  may  be  neglected  only  in 
the  case  of  a  long  bar.  An  excellent  method  of  avoiding  this  dis- 
turbing  influence  of  a  magnetized  piece  of  iron  consists  in  using 
a  ring  instead  of  a  bar  and  in  producing  the  field  by  means  of  a 
solenoid  wound  around  this  ring.  Since  there  are  no  ends  in  this 
case  the  magnetizing  field  is  exactly  equal  to  the  field  calculated 
from  the  equation  (224) 


H  — 


10 


1  = 


10  L 


I 


where  N  is  the  total  number  of  turns  of  wire  and  L  the  length  of 
the  ring  shaped  solenoid  which  is  equal  to  2-n-  times  the  average 
radius  of  the  ring. 

The  magnetic  permeability  ^  of  iron  is  the  ratio  between  the 


*For  a  more  complete  treatment  of  the  properties  of  ferromagnetic 
substances  and  a  description  of  the  various  methods  of  measurement,  see 
J.  A.  Ewing,  Magnetic  Induction  in  Iron  and  other  Metals. 


MAGNETIZATION   OF   IRON 


2O I 


induction  produced  and  the  magnetizing  intensity  of  the  field,  or 

B 
*=  JT 


(244) 


As  shown  in  exercise  79  this  ratio  depends  greatly  upon  the  pre- 
vious history  of  the  piece  under  experiment.  In  order  that  the 
permeability  of  a  substance  shall  give  definite  information  con- 
cerning its  magnetic  quality,  it  has  been  agreed  to  express  the 
permeability  of  iron  and  similar  substances  as  the  above  ratio 
when  the  substance  is  originally  unmagnetized  and  then  subjected 
to  an  increasing  magnetizing  intensity,  (Exercise  78).  Even  un- 
der these  conditions  the  permeability  of  a  given  sample  of  iron 
is  not  a  constant  but  varies  with  the  magnetizing  intensity. 

Under  the  influence  of  variable  or  alternating  magnetic  fields 
iron  and  steel  show  what  is  termed  magnetic  hysteresis,  and  the 
hysteresis  curve  (Exercise  79)  should  be  determined  for  all 
material  which  is  intended  to  be  used  in  alternating  current 
apparatus,  for  example  in  alternators  or  transformers. 

All  substances  which  have  a  large  permeability  approaching  in 
magnitude  that  of  iron  are  called  ferromagnetic  substances. 

153.  Exercise  78.  Commutation  Curve  for  Iron  and  Steel, 
[t  is  of  interest  to  deter- 
mine the  dependence  of 
B  upon  the  intensity  of 
the  magnetizing  field 
produced  by  the  current 
alone,  when  the  intensity 
is  slowly  increased  from 
o  to  30  or  40  gauss.  It 
will  be  seen  from  Fig. 
113,  that  B  increases 
slowly  at  first,  then  very 
rapidly,  then  slowly 
again,  until  it  finally  in- 
creases at  the  same  rate 
as  HI  that  is,  the  pres- 
ence of  the  iron  does  not  introduce  any  additional  lines  of  induc- 
tion. The  iron  is  then  said  to  be  saturated. 


12000 


&ooo 


4000 


2O2 


PHYSICAL    MEASUREMENTS 


The  method  here  described  is  known  as  the  ballistic  or  ring 
method  and  has  the  advantage,  that  the  magnetic  circuit  con- 
tains no  air  gaps  which  offer  large 
magnetic  resistance  and  therefore  ex- 
ert strong  demagnetizing  effects.  (Ar- 
ticle 152).  The  metal  is  given  the 
form  of  a  ring  of  uniform  cross  sec- 
tion, Fig.  114.  It  is  best  to  have  the 
cross  section  of  the  ring  approach  the 
form  of  a  rectangle  and  the  diameter 
large  as  compared  with  the  thickness. 
Fig.  114.  The  outer  and  inner  diameters  of  the 

ring  must  be  measured  with  care,  and  the  volume  of  the  ring 
obtained  from  its  loss  of  weight  in  water.  The  cross  section 
A  is  readily  found  from  the  volume  by  dividing  by  the  average 
circumference.  The  ring  is  then  covered  with  insulating  tape 
upon  which  is  wound  a  layer  of  wire  covering  the  entire  ring 
and  forming  the  primary  coil.  The  number  of  turns  Nif  in  the 
primary  coil  must  be  carefully  determined.  Over  this  is  wound  the 
secondary  coil,  five  to  twenty  turns,  N2.  The  apparatus  should 
be  arranged  as  shown 
in  Fig.  115.  In  the 
primary  circuit  are 
joined  in  series  a  stor- 
age battery,  an  am- 
meter, a  rheostat  R, 
the  primary  coil  of 
the  ring  and  a  com- 
mutator for  reversing 
the  current  through 
the  coil.  The  second- 
ary coil  is  connected 
to  a  ballistic  gal- 
vanometer whose 
constant  and  appar- 
ent resistance  are  FiS-  :I5- 
known  (see  exercise  75).  It  is  most  convenient  to  make  the  total 
galvanometer  resistance  the  critical  resistance  (see  page  142). 


MAGNETIZATION   OF  IRON  203 

The  constant  may  also  be  determined  by   means   of  a  known 

mutual  inductance  M  ;  in  this  case  c  =  —  —  -  ,  where  d  is  the  de- 

ars 

flection  produced  in  the  secondary  by  closing  the  current  /  in  the 
primary.  In  this  case,  however,  the  total  resistance  of  the  second- 
ary must  remain  the  same,  either  by  keeping  the  secondary  of 
the  induction  coil  in  the  circuit  or  by  substituting  an  equal  resist- 
ance for  it.  What  formula  must  then  be  used  instead  of  (249)  ? 
What  is  the  numerical  factor,  if  M  be  expressed  in  millihenry  s 
(10°  c.  g.  s.  units)  and  /  in  amperes  (lO"1  c.  g.  s.  units)  ? 

If  we  vary  the  number  of  lines  of  magnetic  induction  in  the 
solenoid,  the  electromotive  force  induced  in  each  turn  of  the  sec- 
ondary is  at  any  moment'  equal  to  the  rate  of  change  of  the  flux 
threading  through  the  circuit,  or 


If  there  are  A\  turns  of  wire  in  the  secondary  coil  then 

e  =  -N*'7T'  (246) 

The  current  i,  passing  through  the  galvanometer  is  then 

,-  =  -  *    £  (247) 

r2     d  t 

where  rz  is  the  total  resistance  of  the  secondary  circuit.  The 
total  quantity  of  electricity  Q,  passing  through  the  ballistic  gal- 
vanometer is  Cidt,  corresponding  to  a  change  of  $  lines  of 
induction,  or 


where  c  is  the  constant  of  the  galvanometer  and  d  the  throw. 
Express  c  in  c.  g.  s.  units.  Since  the  constant  is  usually  given 
in  micro-coulombs  per  scale  unit,  one  micro-coulomb  being  io~7 


2O4  PHYSICAL    MEASUREMENTS 

c.  g.  s.  unit,  and  the  resistance  in  ohms,  one  ohm  being  equal  to 
10°  c.  g.  s.  units,  the  formula  becomes 


*  =  IO°-    jvT'-  (249) 

Since  B  denotes  the  number  of  lines  of  induction  per  unit  area 
the  increase  of  B  corresponding  to  a  deflection  d  of  the  galvanome- 
ter when  the  current  is  varied  in  the  primary,  is  given  by 


(250) 


To  obtain  the  commutation  curve  it  is  important  to  start  with 
an  unmagnetized  ring.  It  is  best  to  demagnetize  the  ring  by 
sending  first  a  rather  large  current,  say  three  amperes,  through 
the  primary,  increasing  the  resistance  R,  and  then  reversing  the 
direction  of  the  current.  This  must  be  done  several  times,  tak- 
ing care  to  decrease  the  current  each  time  before  the  reversal, 
until  the  current  has  become  very  small.  On  breaking  the  cir- 
cuit the  ring  will  contain  no  residual  magnetism.  A  more  con- 
venient way  is  to  connect  the  primary  to  the  terminals  of  a  small 
alternator  driven  by  a  belt.  Let  the  alternator  attain  its  full  speed 
and  then  throw  off  the  belt.  As  the  speed  decreases  the  current 
decreases,  and  the  iron  is  as  before  subjected  to  cycles  of  con- 
stantly decreasing  magnetic  intensity. 

Arrange  the  apparatus,  as  shown  in  Fig.  115.  Adjust  the  re- 
sistance so  as  to  give  a  small  current  through  the  primary  coil. 
Reverse  the  current  and  observe  the  first  ballistic  throw  of  the 
needle  d\  reverse  again  and  observe  d'\  ;  the  mean,  rfly  corre- 
sponds to  a  reversal  of  the  current  /,  read  from  the  ammeter. 
This  is  repeated,  increasing  the  current  step  by  step,  until  the 
deflections  increase  very  slowly  with  increasing  current.  Compute 
H  from  equation  (224),  and  B  from  (250),  remembering  that  but 
half  the  average  deflection  must  be  taken,  since  the  current 
is  changed  each  time  by  2.1  instead  of  by  /. 

Plot  B  and  H  ',  the  curve  will  resemble  that  shown  in  Fig. 
113.  From  the  curve  it  appears  at  once  that  /JL  is  not  a  constant, 
but  varies  greatly  with  the  degree  of  magnetization. 


MAGNETIZATION   OF  IRON 


205 


FORM   OF   RECORD. 

Exercise  78.     Commutation  curve  for  iron. 


Galvanometer 

Constant  of  galvanometer 

Ring Outer  diam 

Volume  of  ring /        d' 

Crosssection 

Turns  in  primary 

Turns  in  secondary 


Date  

Res 
Inne 
d" 

Ammeter  I1 
istance  r2  

To  

Ave 
H 

rage 

d 

d/2 

B 

A* 

154.  Exercise  79.  Hysteresis  Curve  for  Iron  or  Steel. 
If  instead  of  reversing  the  current  in  the  previous  exercise  we 
should  begin  with  a  large  current,  decrease  /  by  small  steps,  ob- 
serve the  corresponding  throws  of  the  galvanometer  and  plot  B 
with  respect  to  H,  we  obtain  an  entirely  new  curve  owing 
to  the  effects  of 
residual  magnetism 
in  the  iron  or  steel. 
The  experiment  thus 
allows  us  to  study  the 
lagging  of  the  in- 
duced magnetism  be- 
hind the  magnetizing 
field  intensity.  Thus 
if,  in  the  beginning, 
the  iron  were  well 
magnetized,  the  start- 
ing point  on  the  curve 
would  be  the  highest 
point  to  the  right, 
(Fig.  116),  and  on  re- 
ducing the  current  to 
zero,  there  would  still 
remain  in  the  iron  the 
lines  of  induction  re- 
presented by  the  posi- 
tive ordinate.  The  Fig-  116. 
iron  is  still  magnetized,  but  on  reversing  the  current  the  value  of 
B  decreases  very  rapidly,  becomes  zero  for  a  small  negative  cur- 


2O6 


PHYSICAL,    MEASUREMENTS 


rent,  and  finally  reaches  a  value  symmetrical  to  that  at  the  start- 
ing point,  when  the  current  in  the  negative  direction  reaches  the 
same  value  it.  had  at  the  beginning  in  the  positive  direction. 

On  decreasing  the  current  again  to  zero  and  increasing  it  in 
the  positive  direction  to  the  original  value,  the  curve  will  have  a 
form  resembling  closely  the  first  branch.  The  closed  curve  is 
called  a  hysteresis  curve.  The  value  of  B  for  .zero  current  is 
termed  the  remanence  and  the  value  of  H  for  zero  B  is  called 
the  coercive  force  of  the  iron. 

The  arrangement  of  the  apparatus  is  similar  to  that  used  in 
the  preceding  exercise.  The  commutator  in  the  primary  is  con- 
'nected  as  shown  in  Fig.  117.  Start  the  experiment  with  the  com- 
mutator in  the  position  indicated  by  the  dotted  lines  and  with 
switch  K  closed.  Adjust  rheostat  R,  until  the  largest  current 
to  be  used  in  the  experiment  is  reached.  The  iron  has  then  at- 
tained its  maximum  magnetization,  indicated  by  the  highest 

point  on  the  hysteresis  curve. 

If  now  K  be  opened,  the 

current  is  suddenly  de- 
creased, owing  to  the  addi- 
tion of  resistance  in  rheostat 
Rlt  Adjust  R±  to  differ- 
ent values,  beginning  with 
small  resistance  and  in- 
creasing them  for  every  fol- 
lowing observation.  We  ob- 
tain every  time  on  opening 
K  a  deflection  corre- 
sponding to  the  total  change 
in  the  induction,  due  to  the 
decrease  of  current  to  a 
value  I,  determined  by  the 
resistance  in  R±.  Finally 
the  current  is  broken  by 
making  R±  infinite.  In  this 
way  we  obtain  points  on 

the  hysteresis  curve  corresponding  to  a  decrease  of  the  magnetiz- 
ing field  from  a  maximum  value  to  zero. 


MAGNETIZATION   OF  IRON  2O/ 

To  obtain  the  points  corresponding  to  a  reversal  of  H,  the 
commutator  is  connected  so  as  to  join  a  to  c,  and  a'  to  c' '.  R 
remains  adjusted  for  the  largest  current  and  the  commutator  is 
then  suddenly  reversed.  In  this  way  a  magnetic  field  is  produced  in 
the  iron  in  the  opposite  sense,  the  strength  of  the  field  depend- 
ing- upon  the  resistance  in  R^.  This  resistance  should  now  be 
reduced  from  very  high  values  to  smaller  and  smaller  ones  and 
finally  to  short  circuit.  The  deflections  d  will  now  be  in  l.he 
opposite  direction  to  that  before,  unless  the  galvanometer  or  the 
battery  connection  be  reversed.  It  is  evident  that  by  this  method 
we  obtain  points  for  but  one-half  of  the  hysteresis  curve.  Com- 
plete the  curve  by  drawing  the  other  half.  The  origin  of  the 
axis  of  induction  B,  is  found  by  taking  as  the  ordinate  for  maxi- 
mum induction  the  A  B  corresponding  to  one-half  of  the  maxi- 
mum deflection.  Use  equation  (250). 

FORM    OF   RECORD. 

Exercise  /p.     Hysteresis  curve  for  iron  or  steel. 

Date. . 


Record  apparatus  as  in 
preceding  exercise. 


H 


CHAPTER  X. 

OPTICAL  MEASUREMENTS. 
CURVATURE. 

155.  Curvature  of  Optical  Surfaces.     It  is  shown  in  treatises 
on  optics  that  the  effect  of  a  mirror  or  of  a  lens  of  any  form,  con- 
sists in  impressing  upon  the  wave-front  of  the  luminous  disturb- 
ance a  curvature  directly  related  to  the  curvature  of  the  mir- 
ror or  lens  in  question.     By  definition  the  curvature   at  any 
point  in  a  curve  is  the  reciprocal  of  the  radius  of  the  osculating 
circle  at  that  point.     Since  the  effects  produced  by  mirrors  or 
lenses  are  to  be  predicted  from  a  knowledge  of  their  constants,  it 
becomes  a  matter  of  importance  to  measure  the  curvature  of  an 
optical  surface,  in  other  words  to  determine  its  radius  of  curva- 
ture. 

The  surface  most  commonly  employed  in  optical  construction 
is  that  of  the  sphere  since  only  the  largest  mirrors  or  lenses  pos- 
sess surfaces  differing  noticeably  therefrom.  .Concave  or  convex 
mirrors  may  therefore  be  regarded  as  parts  of  spherical  shells, 
with  the  inner  or  outer  surface  polished  as  the  case  may  be.  The 
radius  of  curvature  of  such  a  mirror  is  obviously  the  radius  of 
the  sphere,  of  which  the  mirror  forms  a  part.  In  lenses  both 
surfaces  are  to  be  regarded  as  parts  of  spheres  of  definite  radii. 
In  the  case  of  a  plane  surface  the  radius  is  of  course  infinite. 

156.  Exercise  80.     Radius  of  Curvature  of  a  Lens  by  the 
Spherometer.     The  experiment  consists  in  determining  the  ra- 
dius of  curvature  from  a  careful  measurement  of  the  amount  by 
which  the  lens  surface  departs  from  a  plane,  i.  e.,  by  measur- 
ing the  sagitta1.     If  we  place  a  spherometer  upon  a  lens  with 
the  three   feet   resting  upon   the   surface   of  the   lens,   we   may 
imagine  a  plane  passed  through  the  lens,  cutting  from  it  a  seg- 


1  Preston,  Theory  of  Light,  p.  80. 


CURVATURE 


209 


A} 


nient  whose  base  is  a  circle  passing  through  the  three  feet 
of  the  instrument.  At  right 
angles  to  the  base  of  this  seg- 
ment stands  the  micrometer 
screw  of  the  spherometer,  and 
by  taking  readings,  first  upon 
a  plane  surface  and  then  upon 
the  lens,  the  sagitta  of  the 
curve,  that  is  the  distance  the 
central  foot  of  the  instrument 
is  above  or  below  the  plane 
containing  the  other  three  feet, 
may  be  accurately  determined. 
Thus  in  Figure  118  is  shown 
in  perspective  the  lens,  the 
spherometer  in  place,  and  the 
imaginary  segment,  ABC2C^. 
Above  is  the  equilateral  triangle 
formed  by  the  three  feet  Clf  C2, 
Co,  while  B  at  the  center,  marks 
the  point  where  the  radius  BO, 
pierces  the  base  of  the  segment. 
Then  BB=s,  is  the  sagitta, 

EC2s=d,  the  distance  from  the  point  of  the  micrometer  screw  to 
the  center  of  any  leg,  and  C^O=BO=R,  the  radius  of  the 
sphere  of  which  the  lens  is  a  part.  By  geometry  we  have 


or 


whence 


BE  (2#  —  BE)  =  EC,' 


rf'4-r 


2S 


(251) 
(252) 

(253) 


The  distance  d  is  usually  measured  once  for  all  on  the  dividing 
engine  or  comparator,  and  is  called  the  constant  of  the  instrument. 
It  may  also  be  determined  in  terms  of  I,  the  length  of  one  side  of 


2io  PHYSICAL  MEASUREMENTS 

the  equilateral  triangle  .formed  by  the  feet  of  the  spherometer  as 
follows :  Press  the  instrument  firmly  upon  a  piece  of  stiff  paper 
until  the  positions  of  the  three  are  left  sharply  defined.  The 
length  of  the  sides  of  the  triangle  may  then  be  accurately  meas- 
ured by  the  vernier  caliper.  Then  from  Fig.  118, 


=   —  (254) 

44  3 

whence  by  substitution 

R=JI^  +  ~r  (255) 

In  practice  the  spherometer  is  first  placed  on  a  piece  of  plate 
glass  and  the  zero  reading  accurately  determined.  It  is  then 
transferred  to  the  lens  and  the  readings  upon  the  lens  are  made, 
care  being  taken  to  prevent  the  feet  from  slipping  off  the  lens. 
The  difference  between  the  zero  and  the  final  readings  gives  the 
value  of  s.  From  the  known  value  of  d,  the  value  of  R  is  at 
once  computed,  or  the  value  of  /  may  be  determined  as  shown 
above  and  the  value  of  R  computed  from  the  equation  (255). 

FORM   OF  RECORD. 

Exercise  80.  To  determine  the  radii  of  curvature  of  -five  lenses, 
by  the  spherometer. 


d  — 

I  — 

Date 

Zero 

Lens  i 

Lens  2 

Lens  3 

Lens  4 

Lens  5 

Mean 

Mean 

Mean 

Mean 

Mean 

Mean 

s 
R 

s 
R 

.? 
R 

s 
R 

.? 
R 

Measure  both  sides  of  each  lens. 

157.     Exercise    81.     Radius    of    Curvature    by    Reflection. 

The  radius  of  curvature  of  a  polished  spherical  surface  may  be 
determined  by  means  of  purely  optical  considerations  if  we  em- 
ploy the  phenomena  and  formulae  relating  to  spherical  mirrors. 
Assume  that  the  convex  spherical  surface  mm'  (Fig.  119),  is 


CURVATURE) 


211 


placed  before  the  telescope  T}  at  a  distance  A,  and  that  it  receives 

light  from  two  brilliant 

objects  L   and  Z/   sym- 

metrically    placed     with 

respect  to  T.    There  will 

be  formed  in  the  mirror 

mm'  '  ,  two  virtual,   erect 

and    diminished    images 

of  the  objects  L  and  L'  '. 


Owing  to  the  inversion 


Fig.  lip. 


of  these  images  by  the  telescope  they  are  seen  inverted  in  T.  A 
small  scale  ss',  placed  in  contact  with  the  lens  enables  the  observer 
to  read  off  directly  the  apparent  distance  ss'  between  the  two 
images.  Now  since  the  rays  from  L  and  L',  after  reflection  at  s 
and  s'  enter  the  telescope  and  seem  to  come  from  the  images  / 
and  /',  the  normals  Cs  and  Cs'  will,  if  produced,  bisect  approx- 
imately the  angles  LsT  and  L's'T,  and  to  the  same  degree  of 
approximation,  PQ  =  I/2LI/  where  P  and  Q  are  respectively 
the  intersections  on  LU  of  Cs  and  Cs'  produced.  Let  ss'=s; 
OT,  the  distance  from  the  lens  to  the  objective  of  telescope, 
-.A;  OC  =  R  and  LL'=L.  Then  from  the  triangles  PQC  and 
ss'C  we  have 


PQ        CT 
ss'   '"  CO 


L/2 
s 


or 


R 

2  As 

L,  —  2S 


(256) 


(257) 


(258) 


In  practice  the  lens  is  held  in  a  clamp  supported  upon  a  tripod 
base,  one  foot  of  which  bears  an  adjusting  screw  for  tilting  the 
lens  about  a  horizontal  axis.  This  foot  should  stand  in  a  line 
parallel  to  the  axis  of  the  telescope,  and  normal  to  the  lens  sur- 
face. Two  small  lamps  are  placed  at  L  and  U  with  their  flames 
turned  edge-wise  to  the  lens.  The  telescope  and  lens  are  set  up 
on  two  tables  at  least  three  meters  apart,  the  lens  facing  the  most 


212  PHYSICAL    MEASUREMENTS 

brightly  lighted  window  in  the  room.  The  telescope  is  focused 
upon  the  lens  surface  until  the  scale  ss'  is  sharply  defined.  One 
observer  then  takes  one  of  the  lamps  and  moves  it  slowly  back 
and  forth  and  up  and  down  along  the  line  TL,  until  the  other 
catches  sight  of  the  moving  image  in  the  telescope. 

It  is  to  be  noted  that  the  image  in  a  convex  mirror  is  erect 
and  is  seen  inverted  owing  to  the  inversion  in  the  telescope ;  this 
inversion  applies  to  the  motions  of  the  lamp  as  well,  so  that  if  the 
light  moves  to  the  right,  the  image  seen  in  the  telescope  moves 
to  the  left  and  vice  versa.  Should  the  image  fail  to  appear  when 
the  above  directions  are  followed,  the  lens  holder  should  be  ro- 
tated slightly  about  its  vertical  axis  until  the  image  appears  in 
the  field.  The  image  is  then  brought  to  the  level  of  the  scale  by 
means  of  the  adjusting  screw  in  the  foot  of  the  lens  holder.  A 
black  cloth  placed  close  behind  the  lens  renders  the  image  much 
more  bright  and  distinct. 

Care  must  be  taken  to  avoid  confusion  of  the  true  images  from 
the  front  surface  of  the  lens,  with  the  pair  of  erect  images  seen 
in  the  telescope  which  are  due  to  reflection  from  the  back  of  the 
lens ;  these  images  are  originally  inverted  owing  to  the  concave 
surface,  and  are  erected  by  the  telescope.  Which  pair  of  images 
must  be  chosen  in  case  of  a  concave  lens  ?  What  change  is  needed 
in  the  formula? 

It  will  usually  be  found  necessary  to  change  the  focus  of  the 
telescope  very  slightly  in  order  to  fix  sharply  the  position  of  the 
image  on  the  scale.  This  difference  in  focus  becomes  the  more 
marked  the  more  nearly  the  lens  surface  approaches  a  plane.  The 
method  is  therefore  best  adapted  to  lenses  of  large  curvature. 
The  small  scale  may  be  dispensed  with  by  pasting  upon  the  lens 
two  strips  of  paper  with  straight  edges,  parallel  and  facing  each 
other.  The  perpendicular  distance  between  the  edges  of  the  strips 
is  then  carefully  measured  with  the  vernier  caliper  and  recorded. 
The  lamps  are  then  so  adjusted  that  their  respective  images  just 
disappear  behind  the  edges  of  the  paper.  The  measured  dis- 
tance is  then  equal  to  s.  The  distance  A  and  L  should  be  meas- 
ured with  a  steel  tape  or  a  long  stick  and  a  metric  rule. 

Measure  by  this  method  the  radii  of  curvature  of  three  lenses. 


CURVATURE 


2I3 


FORM    OF   RECORD. 

Exercise  81.     To  determine  the  radii  of  curvature   of  three 
lenses  by  the  method  of  reflection. 

Date 

Lens  No.  A  L  s  R 


158.     Exercise   82.     Focal   Length   of   Lenses.  x     From  the 
well  known  formula  for  the  focal  length  of  a  lens 

i  i    ,     i 

T  =  T  +  T  (259) 

we  may  deduce  an  important  relation  under  the  condition  that 
the  object  and  image  remain  at  a  fixed  distance,  greater  than  $f, 
from  each  other.  Let  /  be  the  distance  between  the  object  and 
the  screen  upon  which  the  image  is  received.  Then  there  will- 
be  two  positions  of  the  lens  for  which  a  sharp  image  is  projected 
upon  the  screen,  one  near  the  object  giving  an  enlarged  image, 
with  the  lens  at  a  distance  p  from  the  object  and  q  from  the  image ; 
and  another  nearer  the  screen  giving  a  small  but  bright  image. 
In  this  position  the  distances  p  and  q  are  interchanged,  so  that 
now  the  lens  is  at  a  distance  q  from  the  object.  Let  a  be  the 
distance  between  these  two  positions  of  the  lens.  Then 


whence 


P  +  Q  =  I,  and  q  —  p  =  a 

/4-a  /  — 

q  =  — —    and  p  =  — 


substituting  in  (258)  we  have  2 


F-tr 


(260) 


(261) 


The  apparatus  consists  of  an  optical  bench  about  two  meters 
long,  provided  with  a  scale  reading  to  millimeters  and  two  sup- 

1  College  Physics,  Article  458. 

-  Owing  to  the  fact  that  the  distances  p  and  q  are  not  measured  from 
the  same  point,  but  from  the  two  principal  points  of  the  lens,  this  formula 
is  not  strictly  accurate ;  the  error  is,  however,  not  large.  For  the  correction 
due  to  this  approximation,  see  Glazebrook  and  Shaw,  Practical  Physics, 
P.  350. 


214  PHYSICAL    MEASUREMENTS 

ports  to  carry  the  screen  and  the  lens.  The  object  is  placed  at 
the  zero  end  of  the  scale  and  at  a  suitable  height  above  it  so  that 
the  object,  the  center  of  the  lens  and  the  middle  of  the  screen  are 
all  in  the  same  straight  line.  In  a  thin  board  at  the  zero  end  of 
the  scale  is  cut  a  hole  3  cm  in  diameter.  This  is  closed  by  a  piece 
of  ground  glass,  which  is  strongly  lighted  by  an  incandescent  bulb. 
A  watch  hand  of  elaborate  design  placed  against  the  glass  on 
the  side  toward  the  lens,  forms  a  well  defined,  dark  object  upon 
a  bright  field. 

A  rough  approximation  to  the  value  of  /  may  be  obtained  by 
placing  a  piece  of  white  paper  in  front  of  the  object  and  bringing 
the  lens  toward  it,  until  there  is  formed  upon  the  paper  an  image 
of  the  window  bars  opposite,  or  of  the  trees  and  buildings  out- 
side. The  reading  of  the  lens  carrier  gives  at  once  the  approx- 
imate focal  length.  Why  is  not  this  the  true  value  of  /?  The 
screen  should  then  be  placed  at  a  distance  from  the  object  not 
less  than  five  nor  more  than  seven  times  this  rough  value  of  /. 
(Why?) 

The  lens  is  now  shifted  until  a  sharp  image  is  projected  upon 
the  screen.  The  mean  of  five  settings  is  taken  as  the  position  of 
the  lens.  The  second  position  of  the  lens  for  a  sharp  image  is 
then  determined  in  the  same  way.  The  difference  between  these 
mean  values  is  a,  and  this  value  with  its  related  value  of  the  set- 
ting of  the  screen  I,  will,  when  substituted  in  the  formula,  give 
a  value  for  /.  At  least  three  different  settings  of  the  screen  should 
be  used  and  the  mean  of  the  three  values  of  /  returned  as  the 
focal  length  of  the  lens. 

In  the  case  of  a  concave  lens,  there  can,  of  course,  be  no  real 
image.  Therefore,  in  order  to  use  this  method,  it  is  necessary 
to  combine  the  concave  lens  with  a  convex  lens  of  suitable  cur- 
vature, and  determine  first  the  focal  length  of  the  combination 
and  then  the  focal  length  of  the  convex  lens  separately.  If  F 
be  the  focal  length  of  the  combination,  and  /'  that  of  the  auxiliary 
lens  then  the  focal  length  of  the  concave  lens  /,  is  given  by  the 

relation 

T  _     i         i 

T~~T"  F 

or 


CURVATURE 


215 


FORM    OF   RECORD. 

Exercise  82.     To  determine  the  focal  lengths  of  five  different 
lenses. 

Date 

Lens 


Screen 
(0 

Lens 
ist  position 

Lens 
2nd  position 

a 

/'—  a2 

4l 

f 

159.  Exercise  83. 
Lens  Curves.  We 
have  seen  from  Ex- 
ercise 82  that  for 
every  setting  of  the 
screen  there  are  in 
general  two  positions 
of  the  lens  for  which 
a  sharp  image  is  ob- 
tained. If  now  we 
plot  the  settings  of  the 
screen  as  ordinates 
and  the  corresponding 
settings  of  the  lens  as 
abscissae,  we  obtain 
what  is  known  as  the 
lens  curve,  Fig.  120 
From  our  nomencla- 
ture the  equation  of 
the  curve  is 


Focal  length  of  lens  = 


Fig.  120. 


(263) 


Is  this  the  equation  of  an  hyperbola  ?  If  so,  what  are  its  asymp- 
totes? What  is  the  physical  interpretation  of  each?  Use  for  this 
experiment  a  lens  of  about  15  cm  focal  length.  Determine  at 
least  fifteen  separate  positions  of  the  screen  with  their  related 
settings  of  the  lens.  Plot  the  curve  and  draw  the  asymptotes. 


-2l6 


PHYSICAL   MEASUREMENTS 


Take  pains  to  obtain  as  many  as  four  or  five  points  near  the  bend 
of  the  curve,  i.  e.,  where  the  two  images  approach  each  other. 
What  is  the  value  of  y  for  the  lowest  point  of  the  curve  ?  Deter- 
mine the  focal  length  of  the  lens  from  the  curve. 


FORM    OF   RECORD. 


Exercise  83.    Lens  curves. 

Lens 

Screen  (y}  Lens 


Date 
Lens 


Focal  length  = 


Plot  curve. 


MAGNIFYING  POWER. 


1 60.     Exercise   84.     Magnifying   Power   of  the   Telescope. 

The  telescope  in  its  simplest  form  consists  of  two  lenses,  the  ob- 
ject-glass or  objective  L,  a  convex  lens  of  long  focus,  and  the 


Fig.  121, 


eye-piece  U ,  a  short  focus  lens  either  convex  or  concave.  The 
distance  from  the  object  to  the  instrument  is  always  great  as 
compared  with  the  focal  length  of  the  objective  and  the  image 
is  consequently  smaller  than  the  object  in  all  cases.  In  case  the 
eye-piece  is  a  convex  lens,  (Fig.  121),  this  small  image  is  viewed 


MAGNIFYING   POWER  2I/' 

directly  by  the  eye-piece  as  an  object  placed  nearer  the  lens  than 
its  focal  distance.  The  result  is  a  magnified  virtual  image  of  the 
image. 

The  effect  of  a  telescope  is  to  increase  the  visual  angle  sub- 
tended by  a  very  distant  object,  that  is,  to  bring  an  image  of  the 
object  near  the  eye,  so  that  when  this  image  is  viewed  by  the  eye- 
directly,  the  visual  angle  subtended  by  it  is  larger  than  that  sub- 
tended by  the  object,  in  the  ratio  F/25,  where  F  is  the  focal 
length  of  the  objective  and  25  cms  represents  the  distance  of  dis- 
tinct vision  for  the  normal  eye.1  This  relation  is  readily  seen  from 
Fig.  122,  where  the  objec- 
tive L  forms  an  image  of  a 
distant  object  upon  a  screen. 
An  eye  at  the  center  of  the 
objective  would  see  both  ,  Fig.  122. 

image  and  object  as  of  the  same  size,  since  the  subtended  angles 
are  equal.  If  however,  the  eye  approach  the  screen,  the  angle 
subtended  by  the  image  will  increase  until  at  a  distance  of  twenty- 
five  cms  from  the  screen  the  image  will  appear  larger  than  the 
object  in  the  ratio  F/2$,  as  given  above. 

If  the  eye  be  brought  nearer  to  the  image  in  order  to  increase 
the  magnification,  its  power  must  be  increased  by  the  use  of  a 
lens  as  a  simple  magnifier.  Such  a  lens  is  termed  an  eye- 
piece. The  magnification  produced  by  the  eye-piece  is  2$/f  where 
the  focal  length  of  the  eye-piece  is  /.  The  total  magnification  of 
the  two  lenses  forming  the  telescope  is  therefore  the  product  of 
the  two,  or  F/f.  In  case  the  object  is  not  at  a  great  distance,  F 
is  no  longer  the  focal  length  of  the  objective,  but  is  the  distance 
from  the  objective  to  the  image  formed  by  it.  Consequently 
F/f  changes  with  the  distance  of  the  object  from  the  telescope. 
The  ratio  F/f  is  called  the  magnifying  power  of  the  telescope, 
and  is  most  readily  measured  as  follows: 

A  long  scale  is  set  up  at  one  end  of  the  room  and  so  lighted 
that  the  divisions  shall  be  seen  sharp  and  clear.  The  telescope 

1  College  Physics,  Article  476. 


218 


PHYSICAL    MEASUREMENTS 


is  focused  upon  the  scale  so  as  to  give  a  sharp  image.  The  ob- 
server next  looks  through  the  telescope  with  the  right 
eye  and  views  the  scale  directly  with  the  left  eye.  A 
little  adjustment  of  the  direction  of  the  telescope  and 
a  little  patience  will  enable  the  observer  to  see  the  two 
images  formed  by  the  two  eyes,  overlapping,  so  that 
he  sees  at  the  same  time,  (Fig.  123),  the  complete  scale, 
and  projected  upon  it,  the  magnified  divisions  of  the 
scale  image.  By  careful  adjustment  of  the  telescope  the 
lengths  of  these  magnified  divisions  may  be  read  di- 
rectly in  terms  of  the  divisions  of  the  scale.  Thus,  sup- 
J7  pose  the  half  division  from  4  to  4^  is  seen  projected 
upon  the  scale,  its  upper  edge  appearing  to  be  at  4.10 
and  its  lower  edge  at  8.15.  It  is  clear  that  one  half 
g<  123>  division  seems  to  cover  4.05  divisions,  hence  the  mag- 
nifying power  is  8.10. 

Care  should  be  taken  to  avoid  touching  the  telescope  or  its 
support  during  the  measurements,  as  well  as  to  avoid  moving 
the  head  while  comparing  the  upper  and  lower  edges  of  the 
image  for  coincidence  with  the  scale  divisions.  Measure  the 
magnifying  power  of  the  telescope  at  distances  of  4,  7,  10,  and  15 
meters  from  the  scale.  Next  remove  the  field  combination,  by 
unscrewing  the  telescope  at  the  first  joint  from  the  eye-piece,  and 
taking  out  the  lens  found  there.  Repeat  the  measurement  as 
above.  What  is  the  purpose  of  the  field  combination  ?  How  does 
the  magnification  vary  with  the  distance? 

FORM   OF  RECORD. 

Exercise  84.  To  determine  the  magnifying  power  of  a  tele- 
scope. 

Date 

Distance  ffom  scale  Magnifying  power 

with  combination        without  combination 


Effect  of  distance  upon  magnifying  power. 

161.  Exercise  85.  Magnifying  Power  of  Microscope.  In 
a  precisely  similar  way  the  magnification  of  a  compound  micro- 
scope may  be  measured  by  placing  upon  the  stage  of  the  instru- 


MAGNIFYING    POWER 

merit  an  object  micrometer,  usually  one  containing  a  millimeter 
subdivided  into  tenths  and  hundredths,  and  focusing  it  sharply 
with  good  transmitted  illumination.  A  millimeter  scale  held  at 
the  distance  of  distinct  vision  from  the  eye-piece  is  next  placed  in 
position  and  adjusted  until,  on  looking  through  the  microscope 
with  both  eyes  open,  the  magnified  image  is  seen  projected  upon 
the  scale,  and  certain  prominent  lines  of  the  two  scales  are  made 
to  coincide.  The  computation  is  identical  with  that  in  Exercise 

84. 

An  alternative  method  is  to  place  on  the  stage  of  the  microscope 
the  object  micrometer,  and  by  means  of  a  camera  lucida,  or  an 
Abbe  illuminating  prism,  project  the  image  of  the  scale  directly 
into  the  eye-piece  and  view  the  two  images  with  a  single  eye. 

In  the  case  of  instruments  provided  with  a  micrometer  eye-piece 
the  magnification  is  determined  by  measuring  with  the  microm- 
eter the  size  of  the  image  of  known  divisions  on  the  object  scale. 
The  magnification  is  then  the  ratio  between  the  size  of  image  and 
the  size  of  object. 

Most  microscopes  are  provided  with  a  millimeter  scale  on  the 
side  of  the  draw  tube.  Measure  the  magnification  of  the  micro- 
scope for  at  least  three  positions  of  the  tube.  How  does  the  mag- 
nification vary  with  the  position  of  the  eye-piece? 

FORM    01?   RECORD. 

Exercise  85.  To  determine  the  magnifying  pozcer  of  a  micro- 
scope. 

Date 

Microscope  No 

Object  micrometer  No 

Reading  on  draw  tube  Magnifying  power 

1  I 

2  2 

3  3. 

Effect  of  extending  tube 

INDEX  OF  REFRACTION. 

162.  Exercise  86.  Index  of  Refraction  of  Lenses  from 
Radii  of  Curvature  and  Focal  Lengths.  It  is  shown  in  geomet- 
rical optics  that  the  focal  length  of  a  lens  for  any  wave  length  is 
a  function  of  the  index  of  refraction  p,  of  the  glass  for  light  of 


22O 


PHYSICAL    MEASUREMENTS 


that  wave  length,  and  of   the  radii  of  curvature  of  the  lens.     This 
relation  is  given  by  the  equation1 

-j  =  (n-i)  6^-;-~-).  (264) 

If  we  have  the  values  of  /,  r±  and  r2  for  any  lens  we  may  com- 
pute the  index  of  refraction  at  once  from  the  above  formula.  The 
focal  length  having  been  obtained  by  means  of  white  light,  the 
resulting  value  of  /*,  will  of  course  refer  to  no  definite  color,  but 
will  in  general  correspond  to  the  brightest  part  of  the  spectrum, 
i.  e.,  to  the  part  between  the  lines  D  and  £. 

FORM    OF   RECORD. 

Exercise  86.  From  the  values  of  f}  r^  and  r2  for  the  lenses 
measured  in  exercises  80  and  82,  compute  the  mean  index  of  re- 
fraction for  each  lens. 

Date 

Lens  i\  r2  f  ^ 


How  may  equation  (264)  be  simplified,  when  one  of  the  radii  is 
infinite?    When  the  two  radii  are  equal? 

163.     Exercise  87.     Index  of  Refraction  by  Means  of  a  Mi- 

croscope. It  is  shown  in 
works  on  physics2  that  if  a 
point  A,  (Fig.  124),  be  viewed 
vertically  through  a  transpar- 
ent plate  of  thickness  OA 
and  refractive  index  /*,  the 
point  will  appear  to  be  raised 
to  some  position  /  in  the  verti- 
cal, such  that  OA  =  //,  O7,  or 


t 

/ 

1 

'  1 

i 

/ 

A 

Fig.  124. 

where  AI  =  a. 

In  this  way  the  index  of  refraction  of  a  transparent  plate,  or 
of  a.  layer  of  fluid  may  be  determined  by  means  of  a  microscope 
furnished  with  a  scale  and  vernier  on  its  tube. 

1  For  the  derivation  of  this  formula  and  its  interpretation  see  College 
Physics,  Articles  454-457. 

2  College  Physics,  Article  448. 


INDEX  OF  REFRACTION 


221 


In  practice  the  microscope,  fitted  with  a  low  power  objective  is 
focused  upon  a  mark  on  a  piece  of  stiff  paper,  or  better  upon  a 
scratch  in  a  piece  of  flat  metal,  held  upon  the  microscope  stage  by 
means  of  clips  or  bits  of  wax.  The  instrument  having  been 
sharply  focused  upon  some  prominent  feature  of  the  scratch,  the 
position  is  taken  by  reading  the  scale  and  vernier  on  the  tube. 
The  transparent  plate,  usually  a  plate  of  glass  some  5  mm  thick, 
is  next  placed  upon  the  stage  above  and  immediately  in  con- 
tact with  the  scratch  in  the  plate.  The  microscope  is  again 
focused  upon  the  same  feature  of  the  scratch  through  the  plate, 
and  the  reading  taken  as  before.  The  microscope  is  then  focused 
upon  the  upper  surface  of  the  plate  and  the  reading  made.  From 
these  three  readings,  each  being  the  mean  of  at  least  five  separate 
settings,  the  values  of  OA  and  OI  are  readily  determined  and  the 
value  of  /x  computed  from  the  formula. 

For  liquids  a  small  flat-bottomed  dish  is  fastened  to  the  micro- 
scope stage  by  two  bits  of  wax,  and  the  instrument  focused  upon 
a  scratch  on  the  upper  surface  of  the  bottom.  The  liquid  is  added 
by  means  of  a  medicine  dropper  to  a  depth  of  from  3  to  5  mm, 
and  the  reading  taken  upon  the  same  scratch  through  the  liquid. 
A  few  grains  of  lycopodium  powder  are  then  sifted  upon  the 
surface  of  the  liquid,  the  microscope  focused  upon  a  grain  of  the 
floating  powder  and  the  reading  taken  as  before.  For  liquids  the 
instrument  must  of  course  stand  vertical.  In  case  the  readings 
differ  by  as  much  as  0.06  mm,  the  mean  of  a  larger  number  of 
readings  must  be  taken.  The  depth  of  the  liquid  may  be  increased 
after  each  determination,  and  readings  through  the  liquid  and  on 
top  of  the  liquid  give  data  for  a  new  value  of  //,. 

Determine  by  this  method  the  refractive  indices  of  two  pieces 
of  glass  and  of  distilled  water. 

FORM    OF   RECORD. 

Exercise  87.  To  determine  the  indices  of  refraction  of  glass 
and  of  distilled  water  by  means  of  a  microscope. 


Date. 

Reading  on  scratch 

Through  subst. 

On  top 

t 

t—a 

A* 

::::::: 

222  PHYSICAL,    MEASUREMENTS 

THE  SPECTROMETER- 

164.  Description.  A  spectrometer  consists  essentially  of  an 
achromatic  telescope,  a  graduated  circle  and  a  collimator,  or  tel- 
escope with  the  eye-piece  replaced  by  an  adjustable  slit,  for  pro- 
ducing a  beam  of  parallel  rays.  The  telescope  is  mounted  upon 
a  substantial  tripod  base,  so  as  to  move  freely  about  the  axis  of 
rotation  of  the  instrument.  The  collimator  is  usually  fixed  in 
position,  but  has  slight  freedom  of  movement  for  purposes  of  ad- 
justment. The  graduated  circle  may  or  may  not  rotate  about 
the  axis  of  the  instrument,  but  it  must  in  any  case  be  capable  of 
being  firmly  clamped  at  will,  and  be  provided  with  verniers  or 
reading  microscopes  to  determine  the  position  of  the  telescope  at 
any  time. 

In  the  Geneva  Society  instrument  shown  in  Fig.  125,  the  prism 

table  is  provided  with  a 
"  •' •  radial  arm  carrying  a  ver- 
nier along  the  graduated 
circle,  by  means  of  which 
the  rotation  of  the  table  may 
be  accurately  determined. 
By  means  of  a  set  screw  the 
table  may  be  clamped  to 
this  arm  or  released  from  it, 
and  when  free  it  may  be 
raised  or  lowered,  or  rotat- 
Fi£-  I25-  ed  at  will  about  its  own 

axis  for  purposes  of  adjustment.  It  is  also  provided  with  three 
levelling  screws,  symmetrically  placed  about  its  axis  for  bringing 
the  surface  of  prism  or  grating  parallel  with  the  axis  of  rotation. 
Both  the  radial  arm  and  the  arm  carrying  the  telescope  are 
provided  with  clamp  and  slow  motion  screws  for  accurate  setting 
upon  any  desired  feature  in  the  field  of  view.  Through  adjust- 
ing screws  attached  to  their  supports,  both  telescope  and  colli- 
mator are  capable  of  slight  motion  in  a  vertical  plane  in  order  to 
bring  their  optical  axes  accurately  perpendicular  to  the  axis  of 
rotation  of  the  instrument. 

The  telescope  is  also  provided  with   a  Gauss  eye-piece,   for 


THE  SPECTROMETER 


223 


illumination  of  the  cross-hairs  in  making  certain  adjustments  and 
measurements.  This  eye-piece  (Fig.  126),  has  an  opening  in  one 
side  and  a  piece  of  transparent,  plane  parallel  glass  set  across  the 
tube  at  an  angle  of  45°  to  the  axis,  by  which  light  entering  the 
opening  is  reflected  parallel  to  the  optical  axis  of  the  telescope. 
When  the  cross-hairs  are  brought  into  the  focal 
plane  of  the  objective, light  passing  them  leaves 
the  objective  as  parallel  rays  and  after  striking 
a  plane  surface  placed  normal  to  the  axis  of 
the  telescope,  is  reflected  directly  back  into  the 
telescope  forming  in  its  focal  plane  a  dark 
image  of  the  cross-hairs  themselves. 

A  far  more  delicate  and  useful  device  is  the 
Zeiss-Abbe  eye-piece,    (Fig.   127),  in  which 
a  small,  totally  reflecting  prism  is  inserted  in 
the  focal  plane  of  the  telescope  at  one  side  of 
the  field  of  view,  and  provided  with  an  adjust- 
able slit  on  the  side  toward  the  objective.   The  Fig.  126. 
slit  and  cross-hairs  lie  in  the  same  plane,  and  when  the  telescope 
is  focused  for  infinity,  light  entering  through  a  small  window  in 

the  side  of  the  eye-piece  is  reflected 
by  the  prism  through  the  slit  and 
emerges  from  the  objective  as  par- 
allel rays.  A  plane  surface  placed 
normal  to  the  axis  of  the  telescope 
returns  the  light  as  from  an  infinite 
distance  and  shows  a  sharp  image 
of  the  slit  in  the  focal  plane  of  the 
telescope.  By  this  arrangement  the 
functions  of  collimator  and  observ- 
ing telescope  are  combined,  the  re- 
flected image  is  sharp  and  unmis- 
I27-  takable,  and  the  system  offers  many 

important  optical  advantages,  some  of  which  will  be  mentioned 
later. 

165.  Adjustments  of  the  Spectrometer.  Before  the  spectrom- 
eter is  readv  for  use  a  number  of  adjustments  must  be  made, 


224 


PHYSICAL    MEASUREMENTS 


some  of  which  must  be  repeated  at  frequent  intervals,  while  others 
should  be  needed  but  rarely.  It  is  not  expected  that  the  beginner 
should  attempt  such  adjustments  for  himself,  but  rather  that  he 
may  obtain  an  intelligent  idea  of  the  working  of  the  instrument 
from  a  careful  study  of  them.  The  adjustments  are  given  in  the 
order  that  they  should  be  made  by  an  observer  on  beginning 
work  upon  a  new  instrument. 

(a)  The  cross-hairs.     The  eye-piece  has  at  its  focus  a  pair 
of  fine  hairs,  termed  cross-hairs,  which  must  be  sharply  seen  by 
the  eye  on  looking  into  the  telescope.     If  the  cross-hairs  are  not 
sharp,  the  eye-piece  must  either  be  drawn  out  or  pushed  in  with 
a  gentle  twisting  motion  until  they  are  seen  sharply  defined  on  a 
white  field  when  the  telescope  is  turned  toward  the  window.    This 
is  the  first  adjustment  to  be  made  in  every  case,  and  as  the  adjust- 
ment is  slightly  different  for  different  persons,  it  must  be  made 
each  time  before  any  work  is  attempted  with  the  telescope. 

(b)  The  telescope.    The  telescope  must  be  focused  for  parallel 
rays.     In  instruments  provided  with  a  Zeiss-Abbe  eye-piece  this 
adjustment  is  readily  effected  by  placing'  in  front  of  the  objective 
a  piece  of  glass  with  a  good  plain  surface ;  a  piece  of  French  plate 
mirror  glass  will  do  very  well.     After  the  slit  is  illuminated  the 
telescope  is  focused  upon  the  reflected  image  until  it  is  seen  clearly 
defined.    The  telescope  is  then  in  adjustment.    Although  the  same 
end  may  be  accomplished  by  use  of  the  Gauss  eye-piece,  the  result 
is  not  so  satisfactory  since  the  images  are  always  faint  and  lack- 
ing in  distinctive   features  by  which   to  judge  the  accuracy   of 
focus.    A  small  dust  particle  adhering  to  the  hairs  may  sometimes 
furnish  the  desired  criterion. 

(c)  The  collimator.     The  telescope  is  next  turned  so  as  to 
look  directly  into  the  collimator.     A  small  lamp  is  placed  behind 
the  slit,  which  is  slightly  opened,  and  the  outer  end  of  the  colli- 
mator either  pulled  out  or  pushed  in  until  the  slit  is  seen  sharply 
focused  in  the  telescope.    The  images  of  slit  and  cross-hairs  must 
show  no  parallax,  that  is,  there  must  be  no  apparent  motion  of 
slit  and  cross-hairs  with  reference  to  each  other  as  the  head  is 
slightly  moved  from  side  to  side  while  looking  into  the  telescope. 
The  collimator  is  now  in  adjustment,  since  the  telescope  focused 


THE;  SPECTROMETER  225 

for  parallel  rays  shows  the  slit  sharply  defined,  and  the  rays 
emerging  from  the  collimator  objective  must  therefore  be  parallel, 

It  is  well  to  mark  this  position  of  the  collimator  tube  in  order 
that  it  may  be  replaced  with  little  trouble  in  case  it  should  be 
accidentally  displaced.  In  some  instruments  the  collimator  tube 
may  be  clamped  in  position,  once  it  is  adjusted  for  focus,  but  this 
arrangement  is  neither  common  nor  necessary.  For  laboratory 
work  under  ordinary  conditions  it  is  usually  more  expedient  to 
focus  and  adust  the  collimator  beforehand  and  require  the  student 
to  confine  his  manipulations  to  the  telescope  and  prism. 

(d)  The  optic  axes  of  telescope  and  collimator.  i.  The  axes 
of  telescope  and  collimator  must  be  perpendicular  to  the  axis 
of  rotation  of  the  instrument.  This  adjustment  is  most  readily 
secured  by  the  use  of  one  of  the  collimating  eye-pieces  mentioned 
above,  together  with  a  small  plate  of  plane  parallel  glass  silvered 
on  both  sides. 

If  the  Gauss  eye-piece  be  employed,  it  is  inserted  in  the  tele- 
scope and  focused  upon  the  cross-hairs,  care  being  taken  to  leave 
the  glass  reflector  as  nearly  vertical  as  possible.  A  small  lamp  on 
an  adjustable  support  is  then  brought  up  to  within  a  few  inches  of 
the  opening  and  so  adjusted  both  laterally  and  vertically  that  on 
placing  a  square  of  good  mirror  glass  over  the  objective  the 
image  of  the  diaphragm  carrying  the  cross-hairs  is  seen  filled  with 
light,  and  on  focusing  the  telescope  slightly  the  cross-hairs  are 
seen  sharply  defined  on  a  bright  field.  A  slight  tilting  of  the  mir- 
ror glass  will  show  by  the  disappearance  of  the  circle  of  light  that 
the  illuminated  circle  is  due  to  reflection  from  it  and  not  from 
the  posterior  surfaces  of  either  of  the  two  intervening  lenses. 

The  small  plate  of  plane  parallel  glass  is  then  mounted  upon 
the  table  of  the  spectrometer,  either  upon  an  adjustable  table  of 
its  own,  or  upon  two  pieces  of  soft  wax,  so  as  to  stand  as  nearly 
vertical  as  may  be,  and  with  its  face  parallel  to  a  line  connecting 
two  of  the  three  levelling  screws  of  the  prism  table.  It  can  then 
be  rotated  slightly  about  this  line  by  means  of  the  third  screw. 
The  plate  is  then  turned  so  as  to  throw  into  the  telescope  the  light 
issuing  from  the  objective,  and  adjusted  until  the  image  of  the 
cross-hairs  is  again  in  the  field  and  brought  into  coincidence  with 


226  PHYSICAL    MEASUREMENTS 

the  hairs  themselves.  If  on  rotating  the  plate  through  180°  the 
images  are  again  in  coincidence  the  plate  stands  parallel  to  the 
axis  of  rotation,  and  the  axis  of  the  telescope  is  normal  to  it.  If 
this  be  not  the  case  then  the  reflected  image  from  the  second  side 
must  be  brought  into  coincidence  with  the  cross-hairs,  by  altering 
by  equal  amounts  the  positions  of  the  plate  and  telescope.  On 
reversing  the  plate  a  similar  procedure  will  soon  give  the  desired 
adjustment.  Finally  the  telescope  is  turned  to  face  the  collimator, 
the  slit  of  which  has  been  illuminated  and  turned  horizontal.  The 
level  of  the  collimator  is  then  adjusted  until  the  image  of  the  slit 
coincides  with  the  horizontal  cross-hair  of  the  telescope.  If  the 
parallel  plate  be  now  turned  to  reflect  the  image  into  the  telescope, 
the  slit  image  should  remain  on  the  horizontal  hair  as  the  tele- 
scope follows  it  around  the  circle. 

2.  The  optic  axes  must  intersect  the  axis  of  rotation  of  the 
instrument.  This  adjustment  in  many  instruments  is  cared  for 
by  the  maker,  and  the  telescope  and  collimator  are  incapable  of 
motion  about  a  vertical  axis.  In  others  the  two  tubes  are  adjust- 
able about  a  vertical  axis,  and  are  very  liable  to  be  displaced.  The 
adjustment  is  most  readily  secured  by  placing  over  the  objective 
of  the  telescope  a  cap  containing  a  lens  whose  focal  length  is 
slightly  less  than  the  distance  from  objective  to  the  center  of  the 
table.  A  bullet  suspended  by  a  cocoon  fiber  is  then  brought 
directly  over  the  center  of  the  table  by  means  of  a  suitable  sup- 
port, and  the  telescope  turned  about  its  vertical  axis  until  the 
image  of  the  thread  falls  upon  the  intersection  of  the  cross-hairs. 
Its  vertical  axis  is  then  firmly  clamped,  the  lens  removed,  and  the 
telescope  swung  round  facing  the  collimator,  which  is  then  ad- 
justed till  the  slit  image  likewise  falls  in  the  center  of  the  field, 
when  the  collimator  is  also  firmly  clamped. 

The  adjustment  should  not  be  disturbed  except  for  special 
reason,  and  the  student  when  rotating  the  telescope  about  the 
circle,  should  take  hold  close  up  to  the  circle,  and  not  by  the 
outer  end. 

1 66.  Reflecting  Surfaces.  A  reflecting  surface  intended  for 
optical  work  should  be  accurately  ground  and  polished.  Many 
surfaces  polished  on  cloth  show  innumerable  fine  ripples  when 


THE  SPECTROMETER  227 

examined  in  good  light,  at  a  distance  of  about  15  inches  from  the 
eye.  Such  surfaces  are  worthless  for  the  formation  of  optical 
images. 

Surfaces  of  prisms  should  be  true  planes,  standing  normal  to 
the  base  of  the  prism  and  should  extend  up  to  the  edges  of  the 
prism.  Many  otherwise  good  prismatic  surfaces  are  rounded  off 
at  the  edges  by  careless  work  in  polishing.  A  perfect  prism 
surface  should  show  no  curvature  at  the  edges  when  tested  by 
interference  fringes  upon  a  true  plane. 

It  is  frequently  desirable  to  examine  a  reflecting  surface  in 
order  to  see  whether  or  not  it  is  clean,  unbroken  or  properly 
illuminated.  In  many  cases  the  surface  may  be  inaccessible  or  at 
a  temperature  which  renders  close  inspection  impossible.  In  such 
cases  a  telescope  showing  an  image  formed  by  light  reflected  from 
the  surface  in  question  affords  the  means  for  such  examination. 
A  short  focus  lens,  as  a  common  reading  glass,  held  between  the 
telescope  and  the  eye  and  properly  focused,  gives  a  well  denned 
image  of  the  prism  surface,  and  enables  the  operator  to  examine 
it  at  leisure.  This  is  especially  useful  in  work  with  prismatic  and 
interferometer  surfaces,  as  the  exact  condition  of  the  reflecting 
surface  and  the  point  from  which  the  reflected  light  proceeds  can 
be  instantly  and  accurately  determined. 

In  adjusting  the  faces  of  a  prism  for  work  on  the  spectrometer, 
one  face  is  set  at  right  angles  to  the  line  joining  two  of  the  three 
levelling  screws  of  the  table,  in  order  that  its  adjustment  may  be 
undisturbed  by  the  motion  of  the  third  screw.  In  the  case  of  a 
prism  whose  angles  are  each  about  sixty  degrees,  and  on  a 
table  whose  levelling  screws  are  placed  symmetrically  about  the 
axis,  this  arrangement  may  be  realized  for  all  three  faces. 

In  some  prisms  one  face  may  not  be  exactly  perpendicular  to 
the  base  of  the  prism,  and  the  image  of  the  slit  as  reflected  from 
this  side  is  slightly  inclined  to  the  vertical  cross-hair.  In  such 
cases  it  is  impossible  to  have  all  the  faces  parallel  to  the  axis  of 
revolution  at  the  same  time.  This  fault  is  slightly  noticeable  in 
most  prisms,  but  by  bringing  the  image  of  the  slit  to  the  middle 
of  the  field  in  each  case  and  setting  the  cross-hair  upon  the  middle 
of  the  image,  the  error  may  be  rendered  practically  negligible. 


228 


PHYSICAL    MEASUREMENTS 


167.  Exercise  88.  To  Measure  the  Angle  of  a  Prism.  In 
measuring  the  angle  of  a  prism  with  the  spectrometer  shown  in 
Fig.  125,  three  methods  of  procedure  are  possible.  Each  method 
will  be  described  in  turn. 

(a)  Prism  fixed  and  telescope  rotated.  The  spectrometer  hav- 
ing been  put  in  adjustment  the  prism  is  placed  upon  the  table  as 
directed  in  Article  166,  and  its  faces  set  approximately  vertical 
by  the  eye.  The  collimator  slit  is  opened  rather  wide  and  illumi- 
nated by  a  small  lamp.  The  angle  to  be  measured  is  turned  toward 
the  collimator  and  so  placed  as  to  divide  the  opening  of  the  ob- 
jective about  equally.  In  the  case  of  telescopes  of  small  aperture 
it  is  important  also  that  the  angle  be  placed  very  near  the  axis  of 
rotation,  as  otherwise  the  reflected  rays  may  be  thrown  out  of  the 
field  of  view. 

Turn  the  telescope  to  position  T,   (Fig.  128).     A  black  cloth 

placed  loosely  over  the  col- 
limator, prism  and  telescope 
aids  materially  in  finding 
the  reflected  image  of  the 
slit  in  the  telescope.  It  is 
best  to  catch  the  reflected 
image  first  in  the  eye 
placed  close  up  to  the  prism 
and  then,  keeping  the  image 
in  view  slowly  bring  the 
telescope  into  position.  Hav- 
ing brought  the  right  hand 
image  into  the  telescope  ob- 
serve whether  or  not  it  lies 
in  the  center  of  the  field 


I28- 


and  parallel  to  the  vertical  hair.  If  not,  the  prism  must  be  ad- 
justed for  level.  Next  turn  the  telescope  to  position  T'  ,  and  see 
whether  the  left  hand  image  is  also  visible,  whether  it  lies  prop- 
erly in  the  field,  and  whether  the  telescope  is  in  such  a  position 
that  readings  may  be  made  in  each  case. 

If  the  prism  need  adjustment  for  level,  observe  carefully  which 
face  is  most  out,  and  note  the  effect  upon  the  position  of  the 


THE  SPECTROMETER  22Q 

images  when  the  prism  is  slightly  rocked  about  each  edge  forming 
the  angle  in  question.  If  the  prism  has  been  properly  placed  upon 
the  table  a  few  minutes  trial  should  bring  it  into  adjustment.  In 
case  the  faces  are  not  both  normal  to  the  base,  the  images  can- 
not both  be  rendered  vertical.  In  such  case  set  the  intersection 
of  the  cross-wires  on  the  center  of  the  image. 

The  slit  image  is  next  drawn  down*  to  a  narrow  line  by  means 
of  the  adjusting  screw  on  the  collimator,  and  brought  upon  the 
vertical  hair  by  use  of  the  clamp  and  slow  motion  screws.  Owing 
to  the  fact  that  the  eye  can  better  judge  of  the  equality  of  the 
two  bright  parts  of  the  slit  on  either  side  of  the  dark  hair,  it  is 
preferable  not  to  make  the  slit  narrower  than  from  three  to  five 
times  the  width  of  the  hair.  The  setting  having  been  completed 
the  position  of  the  telescope  is  read  and  recorded.  If  the  instru- 
ment have  two  verniers  or  two  microscopes  read  both  each  time 
and  combine  the  readings  as  indicated  in  Article  28.  Unclamp  the 
telescope  and  turn  to  position  Tf  and  repeat  the  operations  just 
described.  Then  return  to  position  T,  set  and  read,  to  make  sure 
that  nothing  has  been  changed.  Combine  this  reading  with  the 
first  made.  The  difference  between  this  mean  and  the  reading  at 
T'  is  twice  the  angle  A,  of  the  prism.  Prove  this.  Displace  the 
prism  slightly  and  repeat  the  measurements  twice.  Take  the 
mean  of  the  three  as  the  final  result. 

(&)  Telescope  fixed,  prism  rotated.  Turn  the  telescope  (Fig. 
129),  so  as  to  make  an  angle  of  20°  to  30°  with 
the  collimator  and  clamp  it.  Rotate  the  prism 
table  and  adjust  until  the  reflected  image  from 
face  AC  enters  the  telescope  and  lies  properly 
in  the  field.  Then  continue  the  rotation  until 
the  image  from  face  AB  also  enters  the  field. 
If  both  images  are  properly  located  for  meas- 
urement, clamp  the  table  to  its  radial  arm,  after 
making  sure  that  the  vernier  can  be  read  in 
each  position.  Bring  the  image  from  the  first 
face  once  more  into  the  field  and  clamp  the  pjg  I2g 

arm,  making  the  final  setting  with  the  slow 
motion  screw.     Record  the  reading.     Unclamp  the  arm,  rotate 


230 


PHYSICAL    MEASUREMENTS 


prism  to  second  position,  set  and  read.  The  angle  through  which 
the  prism  has  been  rotated  is  180°  — A.  Take  three  sets  of  read- 
ings as  under  (a),  unclamping  the  table  from  the  arm  and  displac- 
ing slightly  so  as  to  change  the  readings  in  each  case.  Take  the 
mean  of  the  three  results. 

(c}  By  autocollimation.  The  prism  is  placed  centrally  over 
the  axis%  of  the  table  and  the  telescope,  fitted 
with  either  of  the  collimating  eye-pieces  de- 
scribed in  Article  166,  is  clamped  in  position. 
The  illumination  from  the  side  is  adjusted  and 
tested  by  the  small  mirror  placed  over  the  ob- 
jective. The  prism  is  then  turned  so  that  face 
AC  (Fig.  130),  stands  approximately  normal 
to  the  axis  of  the  telescope  and  slowly  rotated 
until  the  bright  image  either  of  the  diaphragm 
or  slit  enters  the  field.  The  prism  is  then  ad- 
justed until  the  reflected  images  from  the  two 
sides  register  accurately  upon  the  cross-hairs. 
The  table  is  then  clamped  to  the  arm  and  read- 
ings made  upon  the  two  faces  in  succession.  The  angle  through 
which  the  prism  has  been  displaced  is  again  180°-  —  A.  Make 
three  independent  sets  of  readings,  displacing  the  table  relatively 
to  the  arm  after  each  set  as  under  (b). 

FORM   OF  RECORD. 

Exercise  88.    To  measure  the  angle  of  a  prism  by  three  differ- 
ent methods. 


(a)  Vernier    I 
"    '.'    II 


istFace 


Date. 
2nd  Face 


Difference 


Mean  ........ 


(b)   Similar  record. 

(O       " 

Angle  by  method  (a)  = 
"      "          "        (&)  = 


Angle, 


Mean  = 

1 68.    Angles  by  Method  of  Repetition.    In  work  of  precision 
it  is  frequently  of  advantage  to  employ  the  principle  of  repetition 


THE:  SPECTROMETER  231 

in  the  measurement  of  angles.  With  a  spectrometer  in  which  the 
movement  of  telescope  and  of  prism  table  may  be  read  separately, 
or  in  which  the  prism  table  may  move  either  with,  or  independ- 
ently of,  the  graduated  circle,  the  principle  may  be  employed  very 
readily  in  either  method  b  or  c,  as  described  above. 

For  example  suppose  that  in  method  (b),  the  prism  has  been 
rotated  from  the  first  to  the  second  position  by  means  of  the 
radial  arm,  and  the  readings  determined  in  each  case.  If  now 
leaving  the  arm  fixed,  the  table  be  released  and  the  prism  rotated 
back  to  its  first  position,  the  setting  made,  and  the  table  again 
clamped  to  the  arm,  and  arm  and  table  again  rotated  in  the  same 
direction  as  at  first  until  the  slit  image  from  the  second  face  again 
falls  upon  the  cross-hairs,  it  is  clear  that  the  difference  between 
the  initial  and  final  reading  of  the  arm  vernier  will  correspond  to 
a  total  rotation  of  2  (180°  — A)  or  360°  — 2A.  If  the  operation 
be  repeated  once  more  the  displacement  becomes  540°  —  $A,  or, 
in  case  A  is  about  60°,  the  total  displacement  is  about  360°  and 
the  initial  and  final  readings  are  made  upon  the  same  part  of  the 
circle. 

The  method  therefore  consists  in  stepping  off  the  angle  a  suffi- 
cient number  of  times  to  bring  the  index  back  to  the  neighbor- 
hood of  the  starting  point,  where  "the  graduations  are  assumed 
to  differ  little  from  each  other.  It  is  designed  to  eliminate  errors 
of  graduation  in  the  circle,  and  assumes  that  these  errors  are 
greater  than  those  of  individual  settings.  The  method  is  em- 
ployed to  the  greatest  advantage  when  the  angle  to  be  measured 
is  some  aliquot  part  of  360°. 

In  some  instruments  the  graduated  circle  is  read  from  fixed 
verniers  or  microscopes,  and  may  move  either  in  conjunction  with 
the  telescope  or  each  may  move  independently.  In  such  instru- 
ments the  principle  may  be  applied  to  method  (a).  Thus  sup- 
pose the  reading  has  been  made  in  position  T;  the  telescope  is 
clamped  to  the  circle  and  the  two  rotated  into  position  T' ',  the 
prism  remaining  fixed.  The  angle  thus  stepped  off  is  2 A.  The 
circle  is  now  clamped  and  the  telescope  released1,  returned  to  posi- 
tion T,  accurately  set  and  clamped  to  the  circle,  which  is  now  re- 


232  PHYSICAL    MEASUREMENTS 

leased  and  telescope  and  circle  again  moved  forward  throug'h  the 
angle  2 A. 1 

169.  Exercise  89.  Index  of  Refraction  of  a  Glass  Prism. 
The  effect  of  a  prism  upon  light  passing  through  it  is  two-fold. 
The  direction  of  the  light  is  changed,  the  light  being  bent  toward 
the  base  of  the  prism  both  on  entering  and  leaving  the  prism,  and 
secondly,  the  light  is  dispersed  or  broken  up  into  its  constituent 
colors.  If  the  prism  used  in  the  previous  exercise  be  now  placed 
centrally  over  the  center  of  the  table  and  turned  into  the  position 
indicated  by  the  full  line  (Fig.  131),  an  eye,  placed  in  the  position 

indicated  by  the  emergent  light,  will 
perceive  no  longer  a  bright  image  of 
the  slit,  but  a  broad  band  of  color, 
the  spectrum  of  the  light  furnished 
by  the  lamp.  This  spectrum  may 
now  be  received  into  the  telescope 
and  its  parts  examined.  The  best 
effect  is  obtained  by  excluding  all 
stray  light  from  the  telescope  by 
means  of  the  dark  cloth  as  in  Exer- 
cise 88.  By  rotating  the  prism 
slowly  and  following  the  spectrum 
Avith  the  telescope,  a  position  is  soon  found  in  which,  no  matter 
which  way  the  prism  is  rotated,  the  spectrum  conies  to  a  certain 
point  nearest  the  direct  line  from  the  collimator,  stops  and  then 
recedes.  This  is  the  position  of  minimum  deviation.  The  small 
lamp  is  now  removed  and  a  Bunsen  burner  substituted.  The 
burner  is  so  arranged  that  the  colorless  flame  plays  against  the 
tip  of  a  piece  of  asbestos  paper  saturated  with  sodium  nitrate. 
An  intense  yellow  light  results.  On  examining  the  image  in  the 
telescope  it  is  seen  that  the  spectrum  of  this  light  consists  of  a 
single  bright  line,  a  yellow  image  of  the  slit,  the  sodium  spectrum, 
for  this  temperature.  In  spectroscopes  of  high  resolving  power 
this  line  is  readily  seen  to  consist  of  two  lines,  D:  and  ZX. 


1  See  Louis  Bell,  The  Absolute  Wave  Length  of  Light,  Am.  Jour.  Sci, 
XXXV,  pp.  350-352. 


THE  SPECTROMETER  233, 

By  closing  the  openings  of  the  Bunsen  burner  so  as  to  give  the 
luminous  flame,  the  continuous  spectrum  returns  and  we  see 
superposed  upon  it  the  bright  line  due  to  the  vapor  of  incandes- 
cent sodium.  The  non-luminous  flame  having  been  restored,  the 
prism  is  rotated  until  the  position  of  minimum  deviation  for  the 
sodium  line  is  actually  determined,  the  cross-hair  placed  upon 
the  image  of  the  slit,  the  telescope  clamped  and  the  reading  taken. 

The  prism  is  next  rotated  into  the  position  shown  by  the  dotted 
line  in  the  figure.  The  light  is  now  deviated  to  the  left  of  the 
direct  position  and  the  position  of  minimum  deviation  is  deter- 
mined as  before.  The  difference  between  the  two  readings  is  ob- 
viously 2D,  where  D  is  the  angle  of  minimum  deviation  for 
sodium  light.  It  is  shown  in  works  on  physics,1  that  when  the 
prism  is  put  in  the  position  of  minimum  deviation,  the  refractive 
index  /*,  is  defined  by  the  equation 


where  A  is  the  angle  of  the  prism.    Derive  this  formula. 

From  the  measured  values  of  D  and  A  as  obtained  above,  com- 
pute the  value  of  ^  for  sodium  light  for  the  prism  under  experi- 
ment. Repeat  the  experiment  using  lithium  carbonate  in  place 
of  sodium  nitrate. 

FORM   OF  RECORD. 

Exercise  89.  To  determine  the  index  of  refraction  of  a  prism 
for  the  lines  Naa  and  Lia. 

Prism..  Date.. 


r 


r 


2D 


D 


Computation 
log  sin  y2(A  +  D)... 

log  sin  y2  A  

l/2(A-\-D) log /*  ... 

/*  •  •  • 


1  College  Physics,  Article  450. 


234 


PHYSICAL    MEASUREMENTS 


DIFFRACTION. 


170.  Exercise  go.  Wave-length  of  Sodium  Light  by  Dif- 
fraction Grating.  A  Bunsen  burner  (Fig.  132),  is  provided  with 
a  sheet  iron  hood  in -which  is  cut  a  small  triangular  slit,  s,  about 
1.5  cm  long.  Immediately  in  front  and  below  the  slit  is  fixed  a 
meter  rod  held  horizonally  with  the  slit  at  the  center  of  the  rod. 
At  a  distance  of  some  three  or  four  meters  in  front  of  the  slit  is 
placed  the  grating,  held  in  a  suitable  clamp,  with  its  surface  ver- 
tical and  parallel  to  the  meter  rod.  If  the  burner  be  adjusted  for 
the  luminous  flame  the  slit  appears  white  to  the  naked  eye  but 
when  viewed  through  the  grating  the  eye  perceives  in  addition 


Fig.  132. 

to  the  white  slit,  a  number  of  spectra  symmetrically  placed  with 
reference  to  the  central  image.  These  are  diffraction  spectra  and 
are  characterized  (a)  by  the  relative  positions  of  the  various 
colors  with  respect  to  the  slit,  the  violet  being  the  least  diffracted 
and  the  red  the  most;  (b)  by  the  uniformity  of  the  dispersion  of 
the  various  spectra,  each  color  being  seen  at  a  distance  from  the 
slit  directly  proportional  to  the  wave-length  of  the  light  in  ques- 
tion. 

In  practice  the  luminous  flame  is  replaced  by  the  sodium  light 
and  the  colored  spectra  become  a  series  of  yellow  images  of  the 
slit  which,  to  an  eye  placed  behind  the  grating,  are  seen  projected 
upon  the  meter  rod  at  spaces  equidistant  from  the  central  image. 
Beginning  at  the  inner  spectra  measure  carefully  the  distances 
•M'H  between  the  first  two  images  on  either  side  of  the  slit,  s.,s'2, 
the  distance  between  the  next  two,  and  so  on.  Take  half  the 
measured  distance  as  the  distance  of  each  image  from  the  central 
slit,  sslf  ss.2,  and  so  on. 


DIFFRACTION 


235 


If  d  be  the  grating  space,  n  the  order  of  the  spectrum  observed, 
and  0n  the  angle  subtended  at  the  eye  by  the  distance  ssn,  then1 


»  X  —  d  sin 


(267) 


where  A  is  the  wave-length  of  sodium  light.     Hence  for  the  first 
three  or  four  spectra, 


d  sin  6-          d  sin  05 

X  =  d  sin  0i  =  •  =  ,  etc. 


(268) 


In  the  experiment  described  the  distances  ssa.  divided  by  a,  the 
distance  from  the  rod  to  the  grating,  gave  directly  tan  0n,  in  each 
case,  from  which  the  value  of  sin  0n  is  readily  found. 

Determine  by  this  method  the  wave-length  of  sodium  light, 
using  spectra  of  at  least  four  different  orders.  The  value  of  d 
for  the  grating  used  will  be  given  by  the  instructor.  Calculate  A 
in  millimeters. 

FORM    OF   RECORD. 

Exercise  90.  To  measure  the  wave-length  of  sodium  light  by 
diffraction  grating. 

«_>  o 


d—  .... 

s  —  .... 

a  —  .... 

Date  

Su 

S'n  ' 

SnS'n 

SSa 

sin0n 

\ 

2 

a 

n 

171.     Exercise  91.     Constant  of  a  Diffraction  Grating.     If 

in  equation  (267)  the  wave-length  of  light  be  assumed  as  known, 
the  quantities  n  and  sin  0n  may  be  determined  and  d  the  grating 
constant  may  be  computed.  Place  a  transmission  diffraction  grat- 
ing, whose  constant  is  to  be  determined,  centrally  upon  the  table 
of  the  spectrometer  and  set  its  surface  normal  to  the  axis  of  the 
collimator.  This  is  most  readily  done  by  setting  the  telescope 
upon  the  image  of  the  illuminated  collimator  slit,  and  then  set- 


1  College  Physics,  Article  494. 


236 


PHYSICAL    MEASUREMENTS 


ting  the  grating  normal  to  the  axis  of  the  telescope,  by  means 
of  an  auto-collimating  eye-piece. 

Illuminate  the  slit  with  sodium  light,  and  determine  0  for  the 
first  two  spectra  on  either  side  the  central  image.  Assume  for  A. 
its  mean  value  0.0005893  mm,  and  compute  the  value  of  d. 


FORM    OF   RECORD. 


Exercise  pi.     To  determine  the  constant  of  a  diffraction  grat- 
ing by  the  spectrometer. 


Grating  used 
Readings 

First  spectrum 

Second 


Right 


Left 


Date, 


20,=    *i=        d-. 
2  02=    02=        di 
Mean 


172.  Dispersion,  Normal  and  Prismatic.  We  have  seen  in 
Exercise  90,  that  the  spectrum  of  a  diffraction  grating  is  char- 
acterized by  the  uniformity  of  the  dispersion  produced,  the  de- 
viation of  each  color  being  absolutely  fixed  by  the  equation  con- 
necting the  wave-length  of  the  color  in  question  and  the  constant 
of  the  grating.  Spectra  formed  by  two  diffraction  gratings  are 
directly  comparable,  the  ratio  of  their  lengths  being  inversely 
as  the  grating  constants.  Such  spectra  are  termed  normal 
spectra,  and  such  dispersion,  normal  dispersion. 

With  prisms  however  the  case  is  very  different.  If  we  ex- 
amine the  spectra  from  a  number  of  prisms  of  different  materials, 
but  all  having  the  same  refracting  angle,  we  shall  find  that  tlv 
lengths  of  the  spectra  differ  enormously.  This  is  said  to  be  due 
to  the  different  dispersive  powers  of  the  different  prismatic  sub- 
stances. Again  if  the  angles  of  the  prisms  be  so  adjusted  that  the 
resulting  spectra  are  all  of  the  same  length,  we  shall  still  find  that 
the  separation  of  the  colors  in  different  parts  of  the  spectrum  is 
very  different  in  the  different  substances.  There  is  no  definite 
relation  therefore  between  the  change  of  index  of  refraction 
and  change  of  wave-length,  the  dispersion  in  each  case  depend- 
ing upon  the  nature  of  the  refracting  substance. 

This  peculiarity  is  termed  irrationality  of  dispersion. 


DIFFRACTION 


237 


173.  Exercise  92.  Dispersion  Curve  for  a  Prism.  Owing 
to  the  irrationality  of  dispersion,  the  spectra  from  prisms  of  differ- 
ent material  are  not  directly  comparable  with  each  other  and  it 
becomes  necessary  to  investigate  experimentally  the  relation  exist- 
ing between  wave-length  and  index  of  refraction  for  any  given 
prism  before  it  can  be  used  as  an  instrument  for  study  of  un- 
known lines  in  the  spectrum. 

This  is  most  readily  done  by  determining  the  indices  of  refrac- 
tion of  the  prism  for  a  number  of  prominent  well  known  lines  in 
different  parts  of  the  spectrum,  and  plotting  on  co-ordinate  paper 
the  indices  as  ordinates  and  the  corresponding  wave-lengths  as 
abscissae.  Through  the  points  thus  determined  a  smooth  curve 
is  drawn  which  is  termed  the  dispersion  curve,  for  the  prism  in 
question. 

By  means  of  this  curve  the  wave-length  of  an  unknown  line 
may  be  determined  as  soon  as  its  index  of  refraction  with  the 
given  prism  is  known,  or  the  index  for  a  line  of  given  wave- 
length may  be  predicted.  The  curve  may  be  represented  with  a 
fair  degree  of  accuracy  by  Cauchy's  dispersion  formula 


(269) 


where  A  and  B  are  constants  depending  upon  the  nature  of  the 
prism  substance.  From  a  series  of  related  values  of  A  and  p,  we 
may  determine  the  constants  A  and  B. 

Determine  according  to  Exercise  89,  the  indices  of  refraction 
of  a  given  prism  for  the  bright  lines  Ka,  K^  Li^  Naa,  Tl,  Sr6^ 
and  plot  a  curve  with  the  indices  thus  obtained  as  ordinates,  and 
the  corresponding  wave-lengths,  (Table  XIII),  as  abscissae.  Take 
off  from  the  curve  the  indices  of  the  prism  for  the  two  hydrogen 
lines  Ha  and  11^  corresponding  to  the  C  and  F  lines  in  the  solar 
spectrum  using  the  values  for  A  given  in  Table  XIII.  Using  sun- 
light verify  the  indices  by  actual  measurement.  Determine  indices 
to  three  decimals. 


238 


PHYSICAL    MKASUREMENTS 


FORM   OF  RECORD. 

Exercise  92.    To  construct  dispersion  curve  for  Prism  No 

and  determine  from  it  the  indices,  of  refraction  for  two  lines  of 
given  wave-length. 


Date, 


Line 
77 

KR 

D 

P> 

X 

/*  for  Ha 
"   H/3 

From  curve 

Observed 

Difference 

TABLES. 


TABLE  I. — Atomic  Weights  of  Some  Elements. 


Hydrogen 
=  1 

Oxygen 
=  16.0 

Hydrogen 

Oxygen 
=  16.0 

Aluminum  

26  90 

27  1 

Nitrogen 

13  93 

14  04 

Cadmium       .... 

111  55 

112  4 

Platinum 

193  30 

194  80 

Chlorine  

35  18 

35  4 

Potassium 

38  85 

39  15 

Copper  

63.12 

63  6 

Silver 

107  11 

107  93 

Gold.  

195.70 

197  2 

Sodium 

22  88 

23  05 

Lead 

205  35 

206  9 

Sulpliur 

31  82 

32  06 

Mercury  

198.50 

200.0 

Tin  

118  10 

119  00 

Nickel  

58.30 

58.7 

Zinc  

64  91 

65  40 

TABLE  II. — Density  of  Water  at  Different  Temperatures. 


t 

d 

/ 

d 

/ 

d 

0° 

0.99988 

11° 

0.99965 

21° 

0.99806 

1 

0.99993 

12 

0.99955 

22 

0.99784 

2 

0.99997 

13 

0.99943 

2;J 

0.99761 

3 

0.99999 

14 

0.99930 

24 

0.99738 

4 

1.00000 

15 

0.99915 

25 

0.99713 

5 

0.99999 

16 

0.99900 

26 

0.99688 

6 

0.99997 

17 

0.99884 

27 

0.99661 

7 

0.99993 

18 

0.99866 

28 

0.99634 

8 

0.99988 

19 

0.99847 

29 

0.99606 

9 

0.99982 

20 

0.99827 

30 

0.99577 

10 

0.99974 

239 


24O  PHYSICAL     MEASUREMENTS 

TABLE  III. — Density  of  Mercury  at  Different  Temperatures. 


t 

t 

tf 

i    * 

d 

0° 

15.5950 

11° 

13.5679 

21° 

13.5433 

1 

.5925 

12 

.5654 

2-2 

.5408 

2 

.5901 

13 

.5629 

23 

.5384 

0 

.5876 

14 

.5605 

24 

.5359 

4 

.5851 

15 

.5581 

25 

.5335 

-  5 

.5827 

16 

.5556 

26 

.5310 

6 

.5802 

17 

.5531 

27 

.52^6 

7 

.5777 

18 

.5507 

28 

.5261 

8 

.5753 

19 

.5482 

29 

.5237 

9 

.5728 

20 

.5457 

30 

.5212 

10 

.5703 

IV. — Densities  of  Various  Bodies. 


Alcohol  at  20°  C 0.789 

.Aluminium 2.58 

Brass (about)  8.5 

Brick 2.1 

Copper 8 . 92 

Cork 0.24 

Diamond 3.52 

Glass,  common 2.6 

"      heavy  flint 3.7 

Gold 19.3 

IceatO°C 0.916 

Iron,  cast 7.4 


Iron,  wrought 7.86 

Lead 11.3 

Mercury  at  0°  C 13.595 

Nickel 8.9 

Oak 0.8 

Paraffin 0.90 

Pine 0.5 

Platinum 21.50 

Quartz 2.65 

Silver 10.53' 

Tin 7.29 

Zinc..  7.15 


TABLES 


24I 


TABLE:  V. — Reduction  of  Barometer  Readings  to  o°C. 

When  the  height  of  a  mercury  column  has  been  measured  with  a 
brass  scale,  the  length  of  which  is  correct  at  o°C.,  the  tempera- 
ture of  the  barometer  being  £°C.,  the  following  quantity,  corre- 
sponding to  temperature  and  height,  has  to  be  subtracted  from  the 
reading.  (See  equation  35.) 


OBSERVED  HEIGHT  OF  BAROMETER. 

t 

72.0  cm. 

73.0  cm. 

74.0cm. 

75.0  cm. 

76.0  cm. 

77.0  cm. 

10° 

0.117  cm. 

0.118  cm. 

0.120  cm. 

0.122  cm. 

0.123  cm. 

0.125  cm. 

11 

.128 

.130 

.132 

.134 

.135 

.137 

12 

.140 

.142 

.144 

.146 

.148 

.150 

13 

.152 

.154 

.156 

.159 

.160 

.162 

14 

.163 

.166 

.168 

.170 

.172 

.175 

15 

.175 

.177 

.180 

.182 

.185 

.187 

16 

.187 

.189 

.192 

.194 

.197 

.200 

17 

.198 

.201 

.204 

.207 

.209 

.212 

18 

.210 

.213 

.216 

.219 

.222 

.225 

19 

.222 

.225 

.228 

.231 

.234 

.237 

20 

.233 

.237 

.240 

.243 

.246 

.249 

21 

.245 

.248 

.252 

.255 

.259 

.262 

22 

.257 

.260 

.264 

.267 

.271 

.274 

23 

.268 

.272 

.276 

.279 

.283 

.287 

24 

.280 

.284 

.288 

.292 

.295 

.299 

25 

.292 

.296 

.300 

.304 

.308 

.312 

26 

.303 

.307 

.312 

.316 

.320 

.324 

27 

.315 

.319 

.324 

.328 

.332 

.337 

28 

.327 

.331 

.336 

.340 

.345 

.349 

29 

.338 

.343 

.348 

.352 

.357 

.362 

30 

.350 

.355 

.360 

.365 

.369 

.374 

TABLE  VI.  —  Coefficients  of  Elasticity. 

Substance       Volume  Elasticity 

Simple  Rigidity        Young's  Modulus        Velocity 

7i  /r                                                Of 

e 

n                             M                    Sound 

Substance 

Volume  Elasticity 
e 

Simple  Rigidity 
n 

Young's  Modulus 

M 

Velocity 
of 
Sound 

Distilled 
water..  .  . 

10'  *  dynes  per  cm2 
0222 

1011  dynes  per 
cms 

1011  dynes  per  cm? 

105  cms  . 
per  sec  . 

1.45 

Glass 
(flint)  .  .  . 
Brass 

3.47  to   4.15 
10  02  to  10  85 

2.35  to  2.40 
3  44  to  4  03 

5.74  to  6.03 
9  48  to  11  2 

5.0 
34 

Steel  
Iron 

(wroug't) 
Iron  (cast). 
Copper  — 

18.41 

14.56 
9.64 
16.84 

8.19 

7.69 
5.32 
4.40  to  4.47 

20.2   to  24.5 

19.63 
13.49 
11.72  to  12.34 

5.1 

51 
5.0 
3.7 

242  PHYSICAL     MEASUREMENTS 

TABLE  VII. — Viscosity  and  Surface  Tension  of  Liquids  at  20° C. 


Coeff.  of  Viscosity 

Surface  Tension 

Alcohol  

grams  per  cm.  sec. 
0.0125 

dynes  per  cm. 
23 

Ether 

0  0025 

18 

Glycerine           .  .             

8  5 

Mercury      

0  015 

68 

Machine  Oil. 

1.95 

Olive  Oil  

0.225 

35 

Turpentine. 

0  0015 

27 

Water       

0  0102 

73 

Uniform     thin 
length  =  / 


TABLE  VIII.—  Moments  of  Inertia. 
Rod,     axis     through     middle, 


Rectangular  Lamina,  axis  through  center  and 
parallel  to  one  side,  a  and  b  length  of  sides,  a 
the  side  bisected  ...... 

Rectangular  Lamina,  axis  through  center  and 
perpendicular  to  the  plane,  a  and  b  length  of 
sides  ...... 

Rectangular  Parallelepiped,  axis  through  center 
and  perpendicular  to  a  side  ;  a,  b  and  c  length 
of  sides,  axis  perpendicular  to  side  contained 
by  a  and  b  ..... 

Circular  Plate,  axis  through  center  perpendicular 
to  the  plate,  radius  =  r 

Circular  Ring,  axis  through  center  perpen- 
dicular to  plane  of  ring,  outer  radius  =  R, 
inner  radius  =  r 

Right  Cylinder,  axis  the  axis  of  figure,  r  =  ra- 
dius of  section  .  .  . 

Sphere,  axis  any  diameter,  r  =  radius 


I=M 


I=M 


a1--  Ir 


TABLES 


243 


TABLE  IX. — Boiling  Point  of  Water  Under  Different  Barometric 

Pressures. 


.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

§72. 
£73. 
o  74. 

I75- 
I  76. 

277. 

a 

a 

98.49 
98.88 
99.26 
99.63 
100.00 
100.37 

98.53 
98.92 
99.29 
99.67 
100.04 
100.40 

98.57 
98.95 
99.33 
99.70 
100.07 
100.44 

98.61 
98.99 
99.37 
99.74 
100.11 
100.48 

98.65 
99.03 
99.41 
99.78 
100.15 
100.52 

98.69 
99.07 
99.44 
99.82 
100.18 
100.55 

98.72 
99.10 
99.48 
99.85 
100.22 
100.59 

98.76 
99.14 
99.52 
99.89 
100.26 
100.63 

98.80 
99.18 
99.55 
99.93 
100.29 
100.67 

98.84 
99.22 
99.59 
99.96 
100.33 
100.70 

TABLE  X.—Heat  Constants. 


Substance 

Cubical 
Expansion 

Specific 
Heat 

Boiling 
Point 

Heat  of 
Vaporization 

Alcohol  

0.00110 

0.58 

78°.  3 

cal. 

210  ~~ 

Ether 

0  00163 

0  54 

34    9 

90 

Mercury 

0  000181 

0  0332 

356    7 

62 

Turpentine 

0  00094 

0  42 

159    0 

70 

Water  

0  00018 

1 

100    0 

539 

Substance 

Linear 
Expansion 

Specific 
Heat 

Melting 
Point 

Heat  of 
Fusion 

Aluminium      

0  000023 

0  22 

658° 

Brass  

0  000019 

0.093 

900 

Copper.  . 

0.000017 

0.093 

1084 

/~M    KK 

(jrlass 

0  000008 

0  19 

1200 

cal. 

Ice         ... 

0  000106 

0  504 

0 

79.5    g 

Iron  

0  000012 

0  11 

1520 

30 

Lead. 

0  000029 

0  031 

327 

6 

Nickel  

0  000013 

0  11 

1450 

Platinum  

0  000009 

0  17 

1755 

27 

Ouartz,  fused  

0  00000045 

0.19 

2000 

Silver 

0  000019 

0  056 

961 

21 

Tin  

0  000023 

0  054 

232 

13 

Zinc  

0  000029 

0  094 

419 

28 

244 


PHYSICAL     MEASUREMENTS 

TABLE  XL — Vapor  Tension  of  Liquids. 


Temperature 

Alcohol 

Ether 

Mercury 

Water 

°C. 

cm. 

cm. 

cm. 

cm. 

—10 

0.65 

11.4 

0 

1.25 

18.5 

0.00004 

0.46 

10 

2.40 

28.8 

0.00005 

0.92 

20 

4.41 

44.0 

0.0001 

1.75 

30 

7.84 

64.5 

0.0003 

3.18 

40 

13.36 

91.0 

0.0006 

5.49 

50 

22.00 

127.5 

0.0013 

9.20 

60 

35.1 

173 

0.0026 

14.89 

70 

54.1 

229 

0.0051 

23.33 

80 

81.2 

300 

0.0092 

35.54 

90 

119 

384 

0.0162 

52.59 

100 

169 

490 

0.028 

76.00 

110 

236 

607 

0.045 

107.5 

120 

322 

760 

0.075 

149.1 

TABLE  XII. — Index  of  Refraction  for  Sodium  Light. 


Dry  Air,  0°C 
Ice 

1.0003 
1  31 

Glass, 
Light  Crown.  . 

1  515 

Calcite, 
Ordinary  rav  

1.659 

Water,  15°C. 
Water,  20°C. 
Alcohol  
Benzine  

1.3336 
1.3332 
1.362 
1.501 

Light  Flint  .  .  . 
Heavy  Flint.  . 
Carbon  Bisul- 
phide, 20°C.  .. 

1.609 
1.75 

1.628 

Extraordinary  ray 
Quartz, 
Ordinary  ray..  .  . 
Extraordinary  ray 

1.486 

1.554 
1.553 

TABLE  XIII. — Wave  Lengths  of  Lines  in  Solar  Spectrum,  in  Air 
at  20°  C,  Pressure  76  cm.;  ^  =  0.001  mm. 


Line 

Element 

Wave 
Length 

Line 

Element 

Wave 
Length 

Line 

Element 

Wave 
Length 

fi 

A1 

A* 

A 

Atm.  O 

0.7628 

D, 

Na 

0.58900 

e 

Fe 

0.43836 

a 

Atm.  O 

0.7185 

Ei 

Fe,  Ca 

0.52703 

f 

H 

0.43405 

B 

O 

0.68701 

h 

Mg 

0.51837 

G 

Fe,  Ca 

0.43079 

C 

H 

0.65629 

-c 

F? 

0.4*576 

h 

H 

0.41018 

a. 

O 

0.62781 

F 

H 

0.48614 

H 

H,  Ca 

0.39685 

A 

Na 

0.58960 

d 

Fe 

0.46682 

K 

Ca 

0.39337 

TABLES^  245 

Wave  Lengths  of  Lines  in  the  Flame  Spectrum  of  Various  Metals. 


Element 

Wave  Length 

Element 

Wave  Length 

K 
Li 
Na 

0.768 
0.671 
0.589 

Tl 
Sr 
K 

0.535 
0.461 
0.404 

TABLE  XIV. — Electrical  Resistance  of  Metals. 
(a)     Specific  conductivity,  referred  to  mercury. 


Aluminium  (soft) 32.35 

Copper  (pure) 59 

Iron 9.75 

Mercury 1 


Nickel  (soft) 3.14 

Platinum 14.4 

Silver  (soft) 62.6 

Tin...  7 


(b)     Specific  resistance,  in  ohms  cm.  at  18°  C. 


Substance 

Specific 
Resistance 

Temperature 
Coefficient 

Aluminium 

0.321 
0  17 
3.0 
1.2 
4.2 
9.58 
1.0 
1.08 
1.4 
0.16  J 

,  XlO'5 

0.0036 
0.0040 
0.0004 
0.0005 
0.00001 
0.00092 
0.005 
0.0036 
0.0025 
0.0037 

Copper 

German  Silver 

Iron  ... 

Manganin  .... 

Mercury  

Nickel  

Platinum,  pure  

Platinum,  commercial  

Silver 

TABLE  XV. — Electrical  Conductivity  of  Solutions  at  i8°C. 
(a)     Specific  conductivity,  in  ohms'1  cm'1. 


Concentration 

NaCl 

CuSOi 

ZnSO4 

AgN03 

5£ 
10* 
15* 

0.0672 
0.1211 
0.1642 

0.0189 
0.0320 
0.0421 

0.0191 
0.0321 
0.0415 

0.0256 
0.0476 
0.0683 

246  PHYSICAL     MEASUREMENTS 

(b)     Equivalent  conductivity. 


Gramequivalents 
per  Liter 

NaCl 

KCl 

KN03 

iH2SO< 

0.01 

0.1028 

0.1224 

0.1182 

0.308 

0.1 

0.0925 

0.1120 

0.1048 

0.225 

0.5 

0.0809 

0.1024 

0.0892 

0.205 

1.0 

0.0743 

0.0983 

0.0805 

0.198 

TABLE  XVI. — Numbers  Frequently  Required. 


i  cm.  =  0.3937  in. 
i  mile  =  1.6093  km. 


IT  2  =  9.8696. 


I  in.  =  2.540  cm. 
i  km.  =  0.6214  mile, 
log  77  =  0.49715. 
log  TT  2  =  0.99430. 


Base  of  natural  logarithms:  £  =  2.7183. 
log  e  =  0.43429. 
Factor  to  convert  common  into  Naperian  logs  =  2.3026. 

Density  of  dry  air  at  o°C.  under  a  barometric 

pressure  of  76  cms.,  0.001293  gm/cm3. 
Coefficient  of  expansion  of  air  .  .  .  0.00367. 
Velocity  of  sound  in  air  at  o°C.  .  332.4  m/sec. 

i  calorie  =  4.181  X  io7  ergs  for  water  at  20°  C. 
i  atmo.  pressure  =  1.0132  X  io6  dynes/cm2, 
g  at  latitude  45°  and  sea  level  =  980.63  cm/sec2. 
Specific  heat  of  steam  at  constant  pressure  .  0.4776. 
Specific  heat  of  steam  at  constant  volume  .  0.3637. 
Vibration  frequency  of  C3  on  the  scientific 

diatonic  scale,  256. 
Vibration  frequency  of  A3  on  the  musical 

equal-tempered  scale,  435. 


TABLES 


247 


TABLE   XVII. — Squares,    Cubes,   Square    Roots,    Circumferences 
and  Areas  of  Circles. 


n 

IT  ;/ 

1A  T  ;/2 

n- 

** 

v« 

1 

3,1416 

0,7854 

1 

1 

1,0000 

2 

6,2832 

3,1410 

4 

8 

4142 

3 

9,4248 

7,0686 

9 

27 

7321 

4 

12,566 

12,566 

16 

64 

2,0000 

5 

15,708 

19,635 

25 

125 

2361 

6 

18,850 

28,274 

36 

216 

4495 

1 

21,991 

38,485 

49 

343 

6458 

8 

25,133 

50,265 

64 

512 

8284 

9 

28,274 

63,617 

81 

729 

3,0000 

10 

31,416 

78,540 

100 

1000 

1623 

11 

34,557 

95,03 

121 

1331 

3166 

12 

37,699 

113,10 

144 

1728 

4641 

13 

40,841 

132,73 

169 

2197 

6056 

14 

43,982 

153,94 

196 

2744 

7417 

15 

47,124 

170,17 

225 

3375 

8730 

16 

50,265 

201,06 

256 

4096 

4,0000 

17 

53,407 

226,98 

289 

4913 

1231 

18 

56,549 

254,47 

324 

5832 

2426 

19 

59,690 

283,53 

361 

6859 

3589 

20 

62,832 

314,16 

400 

8000 

4721 

21 

65,973 

346,36 

441 

9261 

5826 

22 

69,115 

380,13 

484 

10648 

6904 

23 

72,257 

415,48 

529 

12167 

7958 

24 

75,398 

452,39 

576 

13824 

8990 

25 

78,540 

490,87 

625 

15625 

5,0000 

26 

81,68 

530,93 

676 

17576 

099 

27 

84,82 

572,55 

729 

19683 

196 

28 

87,96 

615,75 

784 

21952 

291 

29 

91,11 

660,52 

841 

24389 

385 

30 

94,25 

706,86 

900 

27000 

477 

31 

97,39 

754,77 

961 

29791 

568 

32 

100,53 

804,25 

1024 

32768 

657 

33 

103,67 

855,30 

1089 

35937 

745 

34 

106,81 

907,92 

1156 

39304 

831 

85 

109,96 

962,11 

1225 

42875 

916 

36 

113,10 

1017,9 

1296 

46656 

6,000 

37 

116,24 

1075,2 

1369 

50fi53 

083 

38 

119,38 

1134,1 

1444 

54872       164 

39 

122,52 

1194,6 

1521 

59319      245 

40 

125,66 

1256,6 

1600 

64000 

325 

41 

128,81 

1320,3 

1681 

68921 

403 

42 

131,95 

1385,4 

1764 

74088 

481 

43 

135,09 

1452,2 

1849 

79507 

5o7 

44 

138,23 

1520,5 

1936 

85184 

633 

45 

141,37 

1590,4 

2025 

91125 

708 

46 

144,51 

1661,9 

2116 

97336 

782 

47 

147,65 

1734,9 

2209 

103823 

856 

48 

150,80 

1809,6 

2304 

110592 

928 

49 

153,94 

1885,7 

2401 

117649 

7,000 

50 

167,08 

1963,5 

2500 

125000 

071 

248 


PHYSICAL     MEASUREMENTS 


TABLE  XVII. — Continued. — Squares,  Cubes,  Square  Roots,  Cir- 
cumferences and  Areas  of  Circles. 


n 

if  n 

Xir«2 

«2 

n* 

v« 

51 

160,22 

•  2042,8 

2601 

132651 

7,141 

52 

163,36 

2123,7 

2704 

140608 

211 

53 

166,50 

2206,2 

2809 

148877 

280 

54 

169,65 

2290,2 

2916 

157464 

348 

55 

172,79 

2375,8 

3025 

166375 

416 

56 

175,93 

2463,0 

3136 

175616 

483 

57 

179,07 

2551,8 

3249 

185193 

550 

58 

182,21 

2642,1 

3364 

195112 

616 

59 

185,35 

2734,0 

3481 

205379 

681 

60 

188,50 

2827,4 

3600 

216000 

746 

61 

191,64 

2922,5 

3721 

226981 

810 

62 

194,78 

3019,1 

3844 

238328 

874 

63 

197,92 

3117,2 

3969 

250047 

937 

64 

201,06 

3217,0 

4096 

262144 

8,000 

65 

204,20 

3318,3 

4225 

274625 

062 

66 

207,35 

3421,2 

4356 

287496 

124 

67 

210,49 

3525,7 

4489 

300763 

185 

68 

213,63 

3631,7 

4624 

314432 

246 

69 

216,77 

3739,3 

4761 

328509 

307 

70 

219,91 

3848,5 

4900 

343000 

367 

71 

223,05 

3959,2 

5041 

357911 

426 

72 

226,19 

4071,5 

5184 

373248 

485 

73 

229,34 

4185,4 

5329 

389017 

544 

74 

232,48 

4300,8 

5476 

405224 

602 

75 

235,62 

4417,9 

5625 

421875 

660 

76 

238,76 

4536,5 

5776 

438976 

718 

77 

241,90 

4656,6 

5929 

456533 

775 

78 

245,04 

4778,4 

6084 

474552 

832 

79 

248,19 

4901,7 

6241 

493039 

888 

80 

251,33 

5026,6 

6400 

512000 

944 

81 

254,47 

5153,0 

6561 

531441 

9,000 

82 

257,61 

5281,0 

6724 

551368 

055 

83 

260,75 

5410,6 

6889 

571787 

110 

84 

263,89 

5541,8 

7056 

592704 

165 

85 

267,04 

5674,5 

7225 

614125 

220 

86 

270,18 

5808,8 

7396 

636056 

274- 

87 

273,32 

5944,7 

7569 

658503 

327 

88 

276,46 

6082,1 

7744 

681472 

381 

89 

279,60 

6221,1 

7921 

704969 

434 

90 

282,74 

6361,7 

8100 

729000 

487 

91 

285,88 

6503,9 

8281 

753571 

539 

92 

289,03 

6647,6 

8464 

778688 

592 

93 

292,17 

6792,9 

8649 

804357 

644 

94 

295,31 

6939,8 

8836 

830584 

695 

95 

298,45 

7088,2 

9025 

857375 

747 

96 

301,59 

7238,2 

9216 

884736 

798 

97 

304,73 

7389,8 

9409 

912673 

849 

98 

307,88 

7543,0 

9604 

941192 

899 

99 

311,02 

7697,7 

9801 

970299 

950 

100 

314,16 

7854,0 

10000 

1000000 

10,000 

TABLES 


249 


TABLE  XVIII. — Trigonometric  Functions. 


Arc 

Sine 

Tangent 

log  arc 

log  sin 

log  tan 

1° 

2 

0,0175 
0349 

(V0175 
0349 

0,0175 
0349 

"2,2419 

5428 

"2»2419 

5428 

"2,2419 
5431 

3 

0524 

0523 

0524 

7190 

7188 

7194 

4 

0698 

0698 

0699 

8439 

8436 

8446 

5 

0873 

0872 

0875 

_9408 

_9403 

_9420 

6 

7 

1047 
1222 

1045 
12K) 

1051 

1228 

U)200 
0870 

L0192 
1)859 

1.0216 
0891 

8 

1396 

1392 

1405 

1450 

1436 

1478 

9 

1571 

1564 

1584 

1961 

1943 

1997 

10 

1745 

1736 

1763 

2419 

2397 

2463 

11 

1920 

1908 

1944 

2833 

2806 

2887 

12 

2094 

2079 

2126 

3210 

3179 

3275 

13 

2269 

2250 

2309 

3558 

3521 

3634 

14 

2443 

2419 

2493 

3879 

3837 

3968 

15 

2618  . 

2588 

2679 

4180 

4130 

4281 

16 

2793 

2756 

2867 

4461 

4403 

4575 

17 

2967 

2924 

30)57 

4723 

4659 

4853 

18 

3142 

3030 

3249 

4972 

4900 

5118 

19 

3316 

3256 

3443 

5206 

5126 

5370 

20 

3491 

3420 

3640 

5429 

5341 

5611 

21 

3665 

3584 

3839 

5641 

5543 

5842 

22 

3840 

3746 

4040 

5843 

5736 

6064 

23 

4014 

3907 

4245 

6036 

5919 

6279 

24 

4189 

4067 

4452 

6221 

6093 

64S6 

25 

4363 

4226 

4663 

6398 

6259 

6687 

26 

4538 

4384 

4877 

6569 

6418 

6882 

27 

4712 

4540 

5095 

6732 

6570 

7072 

28 

4887 

4695 

5317 

6890 

6716 

7257 

29 

5061 

4848 

5543 

7042 

6856 

7438 

30 

5236 

5000 

5774 

7190 

6990 

7614 

31 

5411 

5150 

6009 

7333 

7118 

7788 

32 

5585 

5299 

6249 

7470 

7242 

7958 

33 

5760 

5446 

6494 

7604 

7361 

8125 

34 

5934 

5592 

6745 

7733 

7476 

8290 

35 

6109 

5736 

7002 

7860 

7586 

8452 

'  36 

6283 

5878 

7265 

7982 

7692 

8613 

37 

6458 

6018 

7536 

8101 

7795 

8771 

38 

6632 

6157 

7813 

8216 

7893 

8928 

39 

6807 

6293 

8098 

8330 

7989 

9084 

40 

6981 

6428 

8391 

8439 

8081 

9238 

41 

7156 

6561 

8693 

8546 

8169 

9392 

42 

7330 

6691 

9004 

8651 

8255 

9544 

43 

7505 

6820 

9325 

8753 

8338 

9697 

44 

7679 

6947 

9657 

8853 

8418 

9848 

45 

7854 

7071 

\,0000 

8951 

8495 

0,0000 

250  PHYSICAL     MEASUREMENTS 

TABLE  XVIII. — Continued. — Trigonometric  Functions. 


Arc 

Sine 

Tangent 

log  arc     log  sin 

log  tan 

46° 

0,8029 

0,7193 

1,0355 

~T,9047 

1,8569 

0,0152 

47 

8203 

7314 

0724 

9140 

8641 

0303 

48 

8378 

7431 

1106 

9231 

8711 

0456 

49 

8552 

7547 

1504 

9321 

8778 

0608 

50 

8727 

7660 

1918 

9409 

8843 

0762 

51 

8901 

7771 

2349 

9494 

8905 

0916 

52 

9076 

7880 

2799 

9579 

8965 

1072 

53 

9250 

7986 

3270 

9661 

9023 

1229 

54 

9425 

8090 

3764 

9743 

9080 

1387 

55 

9599 

8192 

4281 

9822 

9134 

1548 

56 

9774 

8290 

4826 

9901 

9186 

1710 

57 

9948 

8387 

5399 

9977 

9236 

1875 

58 

1,0123 

8480 

6003 

0,0053 

9284 

2042 

59 

0297 

8572 

6643 

0127 

9331 

2212 

60 

0472 

8660 

7321 

0200 

9375 

2386 

61 

0647 

8746 

8040 

0272 

9418 

2562 

62 

0821 

8829 

8807 

0343 

9459 

2743 

63 

0996 

8910 

9626 

0412 

9499 

2928 

64 

1170 

8988 

2,0503 

0480 

9537 

3118 

65 

1345 

9063 

1445 

0548 

9573 

3313 

66 

1519 

9135 

2460 

0614 

9607 

3514 

67 

1694 

9205 

3559 

0680 

9640 

3721 

68 

1868 

9272 

4751 

0744 

9672 

3936 

69 

2043 

9336 

6051 

0807 

9702 

4158 

70 

2217 

9397 

7475 

0870 

9730 

4389 

71 

2392 

9455 

9042 

0931 

9757 

4630 

72 

2566 

9511 

3,0777 

0992 

9782 

4882 

73 

2741 

9563 

2709 

1052 

9806 

5147 

74 

2915 

9613 

4874 

1111 

9828 

5425 

75 

3090 

9659 

7321 

1169 

9849 

5719 

76 

3265 

9703 

4,0108 

1227 

9869 

6032 

77 

3439 

9744 

3315 

1284 

9887 

6366 

78 

3614  " 

9781 

7046 

1340 

9904 

6725 

79 

3788 

9816 

5,1446 

1395 

9919 

7113 

80 

3963 

9848 

6713 

1450 

9934 

7537 

81 

4137 

9877 

6,3138 

1504 

9946 

8003 

82 

4312 

9903 

7,1154 

1557 

9958 

8522 

83 

4486 

9925 

8,1443 

1609 

9968 

9109 

84 

4661 

9945 

9,5144 

1662 

9976 

9784 

85 

4835 

9962 

11,4301 

1713 

9983 

1,0580 

86 

5010 

9976 

14,3007 

1764 

9989 

1554 

87 

5184 

9986 

19,0811 

1814 

9994 

2806 

88 

5359 

9994 

28,6363 

1864 

9997 

4569 

89 

5533 

9998 

57,2900 

1913 

9999 

7581 

90 

5708 

1,0000 

00 

1961 

0,0000     oo 

TABLES 


251 


XIX. — Logarithms. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

11 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

OS99 

0934 

0969 

1004 

1038 

1072 

1106 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405, 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6(518 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7C67 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

/574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

PHYSICAL,     MEASUREMENTS 

TABLE  XIX. — Continued. — Logarithms. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8.014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306- 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85  ' 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

100 

00000 

0043 

0087 

0130 

0173 

0217 

0260 

0303 

0346 

0389 

101 

00432 

0475 

0518 

0561 

0604 

0647 

0689 

0732 

0775 

0817 

102 

00860 

0903 

0945 

0988 

1030 

1072 

1115 

1157 

1199 

1242 

103 

01284 

1326 

1368 

1410 

1452 

1494 

1536 

1578 

1620 

1662 

104 

01703 

1745 

1787 

1828 

1870 

1912 

1953 

1995 

2036 

2078 

105 

02119 

2160 

2202 

2243 

2284 

2325 

2366 

•^407 

2449 

2490 

106 

02531 

2572 

2612 

2653 

2694 

2735 

2776 

2816 

2857 

2S98 

107 

02938 

2979 

3U19 

3060 

3100 

3141 

3181 

3222 

3262 

3302 

108 

03342 

3383 

3423 

3463 

3503 

3543 

3583 

3623 

3663 

3703 

109 

03743 

3782 

3822 

3862 

3902 

3941 

3981 

4021 

4060 

4100 

110 

04139 

4179 

4218 

4258 

4297 

4336 

4376 

4415 

4454 

4493 

1             !!     1     1     ! 

INDEX 


Numbers   refer   to   Articles 


Abbe     eye-piece     164 

Absorption,   electric    145 

After  effect,    elastic    47 

Air,  expansion  of   Table  XVI 

Air,  free  water 86 

Air    thermometer     87 

Ammeter    116 

Ammeter,    calibration    of    144 

Ampere,    definition    of     100 

Angle,  measurement  of    24-35 

Angle    of   prism,    measurement    of...    167 

Angles  by  repetition 1 68 

Angular  deflection  of  mirror 29 

Astatic  pair  of  magnets    107 

Atomic    weights    Table    I 

Autocollimation,   angle  by    167 

Balance 37-42 

Balance,    Jolly's    54,  71 

Ballistic   galvanometer    113-115 

Ballistic  method   for  magnetic  meas- 
urements        153 

Barometer    46 

Barometer  readings,  corrections  of . .     46 
Batteries,    E.    M.    F.    and    resistance 

of    135-139 

Batteries  of  constant  E.  M.  F 101 

Batteries,   storage    101 

B.  A.  unit  of  resistance    99 

Bending,    laws   of    56 

Boiling    point,    determination    of 80 

Boiling    point    of    water    at    different 

pressures Table  VIII 

Boyle's  law   63 

Buoyancy  of  air,  correction  for    ....     41 

Cadmium     standard     cells     101 

Calibration  of  galvanometer 1 1 1 

Calibration  of  thermometers 80 

Caliper,   micrometer    14 

Caliper,   vernier    18 

Calorie    88 

Calorimeter     90 

Calorimetry      88-95 

Capacity,    comparison    of    145,  146 

Capacity,   definition   of   electrical    .  . .  103 

Capacity,   resistance    128 

Capacity,  standards  of   . . . 103 

Capacity,   thermal    88 

Cadmium  standard  cell    101 

Cathetometer    20 

Cells,    Daniell     101 

Cells,    standard    101 

Cells,    storage    101 

Cells,   in   series   and  in   parallel    ....  117 

Charge  and  discharge  key    115 

Circular    measure    25 

Clock  circuit  for  time  measurement.  44 

Coefficient    of    cubical    expansion    ...  84 
Coefficients  of  elasticity,   Table  VI   49-52 

Coefficients    of   inductance    104 

Coefficient  of  rigidity   52,  58,  59 

Coercive  force    154 

Coincidences,  method  of   44 


Collimator     164,  165 

Commutation   curve,    magnetic    153 

Commutator,    Pohl's     105 

Computation,  hints  on 9 

Condensers,  electric    103 

Conductivity,  electric   128 

Constant   of  ballistic   galvanometer..    114, 

Contact  measurements   1 1 

Controlling   magnet    107 

Copper  coulometer 142 

Coulomb,    definition   of    102 

Coulometer 142 

Current,  measurement  of  electric   140-144 

Curvature   of   lenses    I55-I57 

Curves,    plotting    of     4 

Damping  of  galvanometers    ...".  .108,   112 

Daniell  cell,  treatment  of   101 

D'Arsonyal    galvanometer    ..106,108,112 

Declination,   magnetic    150 

Density,    measurements    of     67-69,54 

Density    ....Tables   II,   III,   IV 

Deviation,    position   of  minimum    . . .    169 
Difference    of    potential,    of    cell    135,136 

Diffraction,    wavelength   by    176 

Diffraction    grating    170,  171 

Diffraction  grating,  constant  of   ....    171 

Dilatometer     85 

Dip,   magnetic 150 

Dispersion    curve    173 

Dividing  engine   22 

Double  weighing   42 

Earth's  magnetic  field   150 

Effective  length  of  bar  magnet    ....    150 
Elasticity,  Coefficients  of,  Table  VI  47-59 

Electricity,    quantity    of    102 

Electrochemical    equivalent    141 

Electrolytic  resistance    128 

E.  M.   F.,  measurement  of   129-138 

E.  M.  F.  of  standard  cells 101 

Equivalent      of      heat,       mechanical, 

Table  XVI 

Errors  of  observation,  influence  of . .        7 

Error,  probable  of  result 6 

Eccentricity,  correction  for   28 

Expansion,    coefficient    of   linear    ..82,83 
Eye-piece   in    telescope    160 

Farad,   definition  of    103 

Figure  of  merit  of  galvanometer 112 

Filar    micrometer,    angles   by    33 

Fixed  points  of  thermometer 80 

Flexure,    Young's   modulus   by    57 

Flux,    magnetic    147,153 

Focal    length    of   lenses,    158 

Freezing  point,   determination  of 80 

Frequency  of  tuning  fork   77 

Fusion,   heat   of,   of   water    93 

g,  measurement  of    62 

Galvanometers     106-108 

Galvanometers,   calibration  of    in 

Galvanometer,    ballistic,    constant    of 


PHYSICAL   MEASUREMENTS 


Galvanometers,    damping  of    108-112 

Galvanometers,  figur  of  merit  of  ...  112 
Galvanometers,  sensitiveness  of  106,  112 

Graphical  methods    4 

Graphical   method    for   tuning   fork..      77 

Grating,    diffraction     170,171 

Gauss   eye-piece    164,    165 

H,    determination   of    150,    151 

Heat   constants, Table    X 

Heat,      mechanical      equivalent      of, 

. Table  XVI 

Heat  of  vaporization    94 

High   resistance,    measurement   of...    122 

Homogeneous     light     169 

Hooke's    law    48 

Hysteresis    curve    154 

Images,    optical    158,     160 

Inclination,  magnetic   150 

Index   of  refraction    Table   XII 

162,  163,  169 

Induction,    magnetic    152 

Inertia,    moments    of    Table    VI 

Instruments,   treatment  of    2 

Insulation  resistance,  measurement  of  122 
Interference  method,  sphereometer  15 
Internal  resistance  of  cells  ....135-139 

Interpolation     8 

Iron    and    steel,    magnetic    properties 
of      . .. 152-154 

Jolly's    balance    54,    71 

Keys,  electrical  105 

KirchhofFs  laws  1 18 

Kohlrausch's  method  for  electrolytic 

resistance  128 

Kundt's  method  for  velocity  of 

sound     75 

Laboratory  work,  benefits  of   i 

Laws   of   bending    56 

Legal   ohm 99 

Length,    measurement    of    11-23 

Lens  curves    159 

Lenses,   curvature   of    1 55-157 

Lenses,  focal  lengths  of    158 

Level,   constants   of    32 

Level   tester 31 

Lever,    optical    30 

Limit    of    elasticity     47 

Line    measurements    19 

Lines   of   magnetic   induction    149 

Magnet,   equivalent  length  of    150 

Magnetic    dip     150 

Megnetic    field     149 

Magnetic  hysteresis    154 

Magnetic    moment    150 

Magnetometer     150 

Magnetisation    curve    153 

Magnifying    power    of    telescope    ...    160 

Mass,  measurement  of   37-42 

Maxwell's     rule     123 

Magnetizing  field    152 

Mechanical       equivalent       of       heat, 

Table  XVI 

Melting  point  of  tin   95 

Melting  points    Table  X 

Mercury,  density  of   Table  III 

Mercury    contacts     105 

Microfarad     103 

Micrometer,  filar   21 

Micrometers,    optical    21 


Micrometer   cathetometer    21 

Micrometer    gauge 14 

Micrometer     screw      13 

Microscope,  index  of  refraction  by..    163 
Microscope,   magnifying  power  of...    161 

Mixtures,    method    of    (heat)     89 

Mixtures,   method   of   (electricity)...    146 

Modulus   of   torsion    52 

Modulus,    Young's. ..  .51,    52,    55,    57,    76 

Mohr's    balance     69 

Molecular    conductivity     128 

Moment   of  couples    52 

Moments  of  inertia   Table  VI 

Moment   of  inertia,   determination   of 

63,  64 
Mutual    inductance     148 

Nernst  and  Haagn's  method  for  bat- 
tery   resistance     139 

Noninductive    winding     99 

Normal     dispersion     172 

Numerical    tables    XV-XVII 

Objective     1 60 

Ohm.    definition    of    99 

Ohms  law,  calibration  by    in 

Optical   lever    30 

Parallax     1 65 

Pendulum,  simple    60 

Pendulum,    torsional     45,    59 

Permeability,    magnetic     158 

Pohl's    commutator    105 

Polarization    of   cells    137 

Post    office    box    124 

Potential    difference   of   cells    ...135,  136 

Potentiometer     132 

Potentiometer  method  for  E.  M.  F..  131 

Prism,    angle    of    167 

Prism,    refraction    through     169 

Prism,    index    of   refraction    of    169 

Prismatic  dispersion    172,  173 

Protractor     68 

Quantity   of   electricity    102 

Radian      25 

Radiation,    correction    for    92 

Radius    of    curvature    iSS^S? 

Record    of    observation    3 

Reflecting    surfaces     166 

Reflection,    radius   of   curvature  by..    157 
Refraction,     index     of,     Table     XII 

162,    163,    169 

Remanence,   definition   of    154 

Repetition,    angle    by     168 

Reports,     final     3 

Resistance    boxes     99 

Resistance   capacity    128 

Resistance,    measurements    of    ...120-128 

Resistance,  standards  of    99 

Resistances,  in  series  and  in  parallel  119 

Resistivity     124 

Resting    point    of    balance     38 

Rheostat     99 

Rigidity,  coefficient  of   52,  58,  59 

Sagitta,     definition     of     is6 

Secohmeter     147 

Selfinductance,    comparison    of    147 

Sclfinductance,    standards    of     104 

Sensitiveness  of  a  balance    39 

Sensitiveness  of  a  galvanometer   106,    112 

Sextant     35 

Shear    5^ 


INDEX 


255 


Shunts      1 10 

Simple    rigidity     52,     58,     59 

Slide    wire    bridge    125 

Sodium   light 169 

Solenoid,   magnetic  field  inside  of   ..    151 

Solids,  expansion  of   Table  X 

Sound,  velocity  of,    Table  VI 

Specific  heat    Table  X  88,  89,  91 

Spectrometer,    The    164,    165 

Spectrometer,    adjustment    of    165 

Spherical  surfaces,  curvatures  of    ...    155 

Spherometer    15,    *S& 

Standard    cells     101 

Standard    resistances    99 

Stem    correction    for    thermometers..     81 

Strain  and  Stress    5  * 

Stretching,   Young's  modulus  by    ...      55 

Subdivision,   method  of    12 

Surface     tension     7*»     72 

Surfaces,    reflecting    166 

Telephone     for     electrical     measure- 
ments     128,  139 

Telescope    and    scale    29 

Telescope,    magnifying    power    of...  160 

Temperature,   definition   of    7° 

Thermal   capacity    88 

Thermo-couple,    E-    M.    F.    of    134 

Thermometer,    calibration    of    80 

Thermometer,  fixed  points  of    80 

Thompson's  method  for  galvanometer 

resistance     •  •  •  126 

Thompson's    method    for    comparing 

capacities     146 

Time,    measurement    of    43.    44 

Torsional     vibrations,     coefficient     of 

simple  rigidity  by    59 

Trigonometric     functions,     angle     by  27 

Tuning    fork,    frequency    of     44,     77 


Units,    fundamental    10 

Units,    electrical     99-104 

Vacuum,  reduction  to  weight  in    ...     41 

Vaporization,  heat  of 94 

Vapor  pressure Table  XI 

Vapor     tension,     measurement     of     96-98 

Velocity    of   sound   in   metals    

Table    VI    75 

Verniers     16 

Vernier   caliper    18 

Vibration,    determining   rate    of    ....      44 

Viscosity     73,     74 

Volt,  definition  of 101 

Voltmeter      116 

Voltmeter,    calibration    of    133 

Water,    air    free    86 

Water,   boiling  point  of    Table  IX 

Water,    vapor   pressure   of    89 

Water   equivalent   of   calorimeter....  90 
Wavelength    of    light,    determination 

of 170 

Wavelength    of    lines    in    solar    spec- 
trum     Table  XIII 

Weighing,  to  make  a  single    40 

Weighing,   double    , .  42 

Weston    standard    cell     101 

Wheatstone    bridge     123 

Wheatstone  bridge  box   124 

Young's   modulus    5 1 

Young's  modulus  by   stretching    ....  55 

Young's    modulus   by   flexure    57 

Young's    modulus,    from    velocity    of 

sound    76 

Zeiss-Abbe    eye-piece    164 

Zero  point  of  thermometers   80 


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